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UNITS AND 
PHYSICAL CONSTANTS. 



UMTS AND 
PHYSICAL CONSTANTS. 



BY 



J. D. EVERETT, M.A., D.C.L., F.R.S., F.R.S.E., 

PROFESSOR OF NATURAL PHILOSOPHY IN QUEEN'S COLLEGE, BELFAST. 



SECOND EDITION. 



MACMILLAN AND CO., 

AND NEW YORK. 

1886. 

[The right of translation and reproduction is reserved. ] 



GLASGOW : PRINTED AT THE UNIVERSITY PRESS 
BY ROBERT MACLEHOSE. 




PREFACE TO "ILLUSTRATIONS OF THE C.G.S. 
SYSTEM OF UNITS," PUBLISHED BY THE 
PHYSICAL SOCIETY OF LONDON IN 1875. 

THE quantitative study of physics, and especially of the 
relations between different branches of physics, is every 
day receiving more attention. 

To facilitate this study, by exemplifying the use of a 
system of units fitted for placing such relations in the 
clearest light, is the main object of the present treatise. 

A complete account is given of the theory of units ab 
initio. The Centimetre-Gramme-Second (or C.G.S.) sys- 
tem is then explained ; and the remainder of the work is 
occupied with illustrations of its application to various 
branches of physics. As a means to this end, the most 
important experimental data relating to each subject are 
concisely presented on one uniform scale a luxury 
hitherto unknown to the scientific calculator. 

I am indebted to several friends for assistance in special 
departments but especially to Professor Clerk Maxwell 
and Professor G. C. Foster, who revised the entire manu- 
script of the work in its original form. 

Great pains have been taken to make the work correct 
as a book of reference. Readers who may discover any 
errors will greatly oblige me by pointing them out. 



PREFACE TO FIRST EDITION OF UNITS 
AND PHYSICAL CONSTANTS, 1879. 

THIS Book is substantially a new edition of my ' ; Illus- 
trations of the C.G.S. System of Units " published in 1875 
by the Physical Society of London, supplemented by an 
extensive collection of physical data. The title has been 
changed with the view of rendering it more generally 
intelligible. 

Additional explanations have been given upon some 
points of theory, especially in connection with Stress and 
Strain, and with Coefficients of Diffusion. Under the 
former head, I have ventured to introduce the terms 
"resilience" and "coefficient of resilience," in order to 
avoid the multiplicity of meanings which have become 
attached to the word "elasticity." 

A still greater innovation has been introduced in an 
extended use of the symbols and processes of multiplication 
and division, in connection with equations which express 
not numerical but physical equality. The advantages of 
this mode of procedure are illustrated by its application 
to the solution of the most difficult problems on units that 
I have been able to collect from standard text-books 
(chiefly from Wormell'a ' Dynamics '). 

I am indebted to several friends for contributions of 
experimental data. 



viii PEEFACE TO FIRST EDITION. 

A Dutch translation of the first edition of this work 
has been made by DR. C. J. MATTHES, Secretary of the 
Royal Academy of Sciences of Amsterdam, and was pub- 
lished in that city in 1877. 

Though the publication is no longer officially connected 
with the Physical Society, the present enlarged edition 
is issued with the Society's full consent and approval. 



PREFACE TO THE PRESENT EDITION. 

IN collecting materials for this edition, I have gone care- 
fully through the Transactions and Proceedings of the 
Royal Society, the Royal Society of Edinburgh, and the 
Physical Society of London, from 1879 onwards, besides, 
consulting numerous papers, both English and foreign, 
which have been sent to me by their authors. I have 
also had the advantage of the co-operation of Dr. Pierre 
Chappuis (of the Bureau International des Poids et 
Mesures), who has for some years been engaged in pre- 
paring a German edition. Several items have been ex- 
tracted from the very elaborate and valuable PJiysikaliscJt- 
Chemische Tabellen of Landolt and Bb'rnstein (Julius 
Springer, Berlin, 1883). Among friends to whom I am 
indebted for data or useful suggestions, are Prof. Barrett, 
Mr. J. T. Bottomley, Prof. G. C. Foster, Prof. Lodge, 
Prof. Newcomb, Mr. Preece, and the Astronomer-Royal. 
The expository portions of the book are for the most 
part unchanged ; but a Supplemental Section has been 
added (p. 34) on physical deductions from the dimensions 
of units ; a simplification has been introduced in the dis- 
cussion of adiabatic compression (p. 125); and the account 
of thermoelectricity (p. 172) has been re-written and 
enlarged. The name "thermoelectric height" has been 
introduced to denote the element usually represented by 
the ordinates of a thermoelectric diagram. 



x PREFACE TO THE PRESENT EDITION. 

The preliminary " Tables for reducing other measures 
to C.G.S. measures" have been greatly extended, and in 
each case the reciprocal factors are given which serve for 
reducing from C.G.S. measures to other measures. Pro- 
fessor Miller's comparison of the kilogramme and pound 
is supplemented by three later comparisons officially made 
at the Bureau International. 

A nearly complete list of the changes and additions 
now introduced is appended to this Preface, as it will 
probably be useful to possessors of the previous edition. 

The adoption of the Centimetre, Gramme, and Second, 
as the fundamental units, by the International Congress 
of Electricians at Paris in 1881, led to the immediate 
execution of a French translation of this work, which was 
published at Paris by Gauthier-Yillars in 1883. The 
German translation was commenced about the same time, 
but the desire to perfect its collection of physical data has 
caused much delay. It will be brought out by Ambrosius 
Earth, the publisher of Wiedemann's Annalen. A Polish 
edition, by Prof. J. J. Boguski, was published at Warsaw 
in 1885 ; and permission has been asked and granted for 
the publication of an Italian edition. 

J. D. EVERETT. 

BELFAST, September, 1886. 



LIST OF CHANGES AND ADDITIONS. 



Tables for conversion to and from C.G.S., 
Formula for g, 

" Watt " defined, 

Physical deductions from dimensions, 
Specific gravity table, - 
Surface tension of liquids, 
Thickness of soap films, - 

Poissoii's ratio, 

Velocity of light, 

Indices of refraction of crystals, etc. , 
Refraction and dispersion of gases, - 

Rotation by quartz, 

Candle, carcel, etc., 

Specific heat, 

Melting, 

Boiling, 

Pressure of steam from to 150, 
Critical points of gases, - 
Conductivity (thermal) of solids, 
,, ,, of liquids, 

Joule's equivalent, 

Adiabatic compression, - 
Expansion of mercury, 
Collected data for air, 

Density of moist air, 

Magnetic susceptibility, ...... 

Greenwich magnetic elements, - 
Magneto-optic rotation, - - - - 

Ratio of the two units of electricity, 
Specific inductive capacity, 



PAGES 

1-4 
26 
30 

- 34-37 

40 

- 49, 50 

- 50, 51 

62 
76 

- 80, 81 

- 83-85 

85 
86 

- 90-94 
95-97 

98 
102 
103 

115, 116 
117 
121 

125, 126 
129 
129 
130 
133 
138 
139 
146 

148-150 



xii CHANGES AND ADDITIONS. 

PAGES 

Practical units, 151, 152 

Resolutions of Congress and Conference, - - - 153, 154 

Resistance, .... 158-164 

Gauge and resistance of wires, - 165, 166 

Electro-motive force of cells, - 167, 168 

Thermoelectricity, - - 172-178 

Electrochemical equivalents, 179-180 

Heat of combination of cells, .... 181, 182 

Compression of liquids, 189 

Expression of decimal multiples, etc., - - - - 190 



CONTENTS. 

PAGES 

Tables for Reducing to and from C.G.S. Measures, - 1-4 

Chapter!. General Theory of Units, - - - 5-18 

-Chapter II. Choice of Three Fundamental Units, - 19-24 

.Chapter III. Mechanical Units, 25-34 

Supplemental Section, on Physical Deductions from 

Dimensions, 34-37 

Chapter IV. -Hydrostatics, - 38-51 

Chapter V. Stress, Strain, and Resilience, - 52-64 

-Chapter VI. Astronomy, - - - 65-69 

Chapter VII. Velocity of Sound, - - 70-74 

Chapter VIII. Light, - 75-86 

Chapter IX. Heat, - 87-130 

. Chapter X. Magnetism, - - - - 131-139 

Chapter XL Electricity, - - - 140-188 

Omission and Suggestion, - - - - - 189, 190 

Appendix. Reports of Units Committee of British 

Association,- ... 191-195 

Index, 196-200 



UNITS AND PHYSICAL CONSTANTS. 



TABLES FOR REDUCING TO AND FROM 
C.G.S. MEASURES. 



The abbreviation cm. is used for centimetre or centimetres, 
gm. gramme or grammes, 

c.c. cubic centimetre(s). 

The numbers headed "reciprocals" are the factors for 
reducing from C.G.S. measures. 

Length. 



1 inch, - 
1 foot, - 
1 yard,- 
1 mile, 
1 nautical mile, - 


cm. 
2-5400 
30-4797 
91-4392 
= 160933 
= 185230 


Reciprocals. 
39370 
032809 
010936 
6-2138 x 10- 6 
5'398xlO- 6 



More exactly, according to Captain Clarke's compari- 
sons of standards of length (printed in 1866), the metre is 
equal to 1-09362311 yard, or 3 -2808693 feet, or 39'370432 
inches, the standard metre being taken as correct at C., 
and the standard yard as correct at 16f C. Hence the 
inch is 2-5399772 centimetres. 

Area. 



1 sq. inch, - 
1 sq. foot, - 
1 sq. yard, - 
1 sq. mile, - 


sq. cm. 
- = 6-4516 
- = 929-01 
- =8361-13 
- = 2-59 x 10 10 
A 


Reciprocals. 
1550 
001076 
0001196 
3-861 x 10-" 



2 UNITS AND PHYSICAL CONSTANTS. 

Volume. 

cub. cm. Reciprocals. 

1 cubic inch, - = 16 '387 '06102 

1 cubic foot, - -28316- 3'532xlO- 5 

1 cubic yard, = 764535' 1 "308 x 10~ 6 

Ipint,- - = 567-63 '001762 

1 gallon, - = 4541' '0002202 

Mass. 

gm. Reciprocals. 

1 grain, - '0647990 15 '432 

1 ounce avoir., - = 28 '3495 -035274 

1 pound - =453-59 '0022046 

1 ton, - - - = 1-01605 xlO 6 9'84206xlO- 7 

According to the comparison made by Professor W. H. 
Miller in 1844 of the "kilogramme des Archives," the 
standard of French weights, with two English pounds of 
platinum, and additional weights, also of platinum, the kilo- 
gramme is 1 5432*34874 grains, of which the new standard 
pound contains 7000. Hence the kilogramme would be 
2-2046212 pounds, and the pound 453*59265 grammes. 

Three standard pounds, one of platinum-indium and 
the other two of gilded bronze, belonging to the Standards 
Department, were compared, in 1883, at the Bureau In- 
ternational des Poids et Mesures, with standards belong- 
ing to the Bureau, and their values in grammes were 
found to be respectively 

453-59135, 

453-58924, 

453-58738. 
Travaux et Memoir es, tome IV. 

Velocity. 

cm. per sec. Reciprocals. 

1 foot per second, - - =30 "4797 '032809 

1 statute mile per hour, - =44 '704 -022369 

1 nautical mile per hour, - =51 -453 '019435 

1 kilometre per hour, - =27 '777 '036 



TABLES. 
Acceleration. 

cm. per sec. per sec. Reciprocal. 
1 ft. per sec. per sec., - =30 '4797 "032809 

Density. 

gm. per c.c. Reciprocals. 



1 lb. per cubic foot, ' - - = "016019 62'426 
1 grain per cubic inch, - = '003954 252 '88 

Stress (in gravitation measure}. 

gm. per sq. cm. Reciprocals. 

1 lb. per sq. foot, - = "48826 2 "0481 
1 lb. per sq. inch, - = 70'31 '014223 

1 inch of mercury at j = 34>534 .^^ 

30 inches ,, ,, = 1036"0 '00096525 

760mm. ,, =1033-3 "00096777 

Surface Tension (in gravitation measure). 

gm. per cm. Reciprocals. 

1 grain per linear inch, - "02551 39 "20 

lib. foot, - =14-88 "06720 

Work (in gravitation measure). 

gm.-cm. Reciprocals. 

1 foot-pound, - =13825 7"2331xlO~ 5 
1 foot-grain, - = 1 -975 "50632 

1 foot-ton, - - = 3'097xl0 7 6 "494 x 10 - & 
1 kilogram metre, - = 10 5 10 ~ 5 

Rate of Working (in gravitation measure). 

gm.-cm. per sec. Reciprocals. 

1 horse-power, - = 7'604xl0 6 l"3151x!0- 7 

1 force-de-cheval, - = 7'SxlO 6 l'3333xlO- 7 

Heat (in gravitation measure). 

gm.-cm. Reciprocals. 

1 gm. deg., - - =42400 2'36xlO- 5 

1 lb. deg. Cent., - = l'923x!0 7 5"2xlO- 8 

1 ,, Fahr., - = l"068x!07 9"36xlQ- 8 

The following reductions of gravitation measures to 
absolute measures are on the assumption that # = 981 : 



UNITS AND PHYSICAL CONSTANTS. 
Force (in absolute measure). 

Dynes. Reciprocals. 

Weight of 1 gm., - - = 981 '001019 

Ikilogm., - = 9-81 xlO 5 1 '019x10-* 

1 tonne, - = 9'81 x 10* l'019x!0- 9 

Iton, . - = 9-97 xlO 8 1'003 x lO' 9 

Icwt.,- - = 4-98 xlO 7 2-008x10-* 

1 Ib. avoir., - = 4'45 x 10 5 2'247 x 10-" 

1 oz. - = 2'78^< 10 4 3-597 x 10- 5 

1 grain, - = 63'57 '01573 

i puandal, =13825 7-2333x10-- 
(The ratio of the poundal to the dyne is independent of g. ) 

Stress (in absolute measure). 

Dynes per sq. cm. Reciprocals 

1 Ib. per sq. foot, - = '479 '00209 

lib. inch, - = 6'9 xlO 4 1 -45xlO- 5 

1 gm. cm., '981 '00102 

1 kilo. decim., - = 9'SlxlO 3 l'02xlO- 4 
1 cm. of mercury at C. , = 13338' '0000736 

76 = 1-0136 xlO 6 9-866 xlO- 7 

1 inch = 3-388 x 10 4 2'95 x 10~ 5 

30 l-0163x!0 6 9-84xlO- 7 

Surface Tension (in absolute measure}. 

Dynes per cm. Reciprocals. 

Igm. per linear cm., = 981 '00102 

1 grain ,, inch, - = 25 '04 

lib. foot, - - =14600 6-85 x 10-* 

Work and Energy (in absolute measure). 

Ergs. Reciprocals. 

1 gm. cm., - - = 981 '001019 

1 kilogrammetre, = 9 '81 x 10 7 1 '019 x 10~ 8 

1 foot-pound, - = 1-356x107 7 '37x10-* 

1 foot-poundal, - =421390 2-3731 x 10~ 6 
(The ratio of the ft. -poundal to the erg is independent of g.) 
] joule - - = 10 7 ergs. 

Rate of Working (in absolute measure). 

Ergs per sec. Reciprocals. 

1 horse-power, - - = 7 '46 x 1 9 1 '34 x 1 - 10 

1 force-de-cheval, - - =7'36xl0 9 l'36xlO- 1( > 

1 watt, - - - - = 10 7 lO- 7 

Heat (in absolute measure). 

Ergs. Reciprocals. 

Igm. deg., - - - =4-2 xlO 7 2'38xlO- 8 

! deg. Cent., - - =l'905x!0 10 S^SxlQ- 11 

Fahr., - - =l'058x 10 10 9-45xlO' n 



1 gm 
I Ib. 



CHAPTER I. 

GENERAL THEORY OF UNITS. 
Units and Derived Units. 

1. THE numerical value of a concrete quantity is its 
ratio to a selected magnitude of the same kind, called 
the unit. 

Thus, if L denote a definite length, and I the unit 

length, - is a ratio in the strict Euclidian sense, and is 

called the numerical value of L. 

The numerical value of a concrete quantity varies 
directly as the concrete quantity itself, and inversely as 
the unit in terms of which it is expressed. 

2. A unit of one kind of quantity is sometimes defined 
by reference to a unit of another kind of quantity. For 
example, the unit of area is commonly defined to be the 
area of the square described upon the unit of length ; 
and the unit of volume is commonly defined as the volume 
of the cube constructed on the unit of length. The units 
of area and volume thus defined are called derived unite, 
and are more convenient for calculation than indepen- 
dent units would be. For example, when the above 



6 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

definition of the unit of area is employed, we can assert 
that [the numerical value of] the area of any rectangle is 
equal to the product of [the numerical values of] its 
length and breadth ; whereas, if any other unit of area 
were employed, we should have to introduce a third factor 
which would be constant for all rectangles. 

3. Still more frequently, a unit of one kind of quantity 
is defined by reference to two or more units of other 
kinds. For example, the unit of velocity is commonly 
defined to be that velocity with which the unit length 
would be described in the unit time. When we specify 
a velocity as so many miles per hour, or so many feet per 
second, we in effect employ as the unit of velocity a mile 
per hour in the former case, and a foot per second in the 
latter. These are derived units of velocity. 

Again, the unit acceleration is commonly defined to 
be that acceleration with which a unit of velocity would 
be gained in a unit of time. The unit of acceleration is 
thus derived directly from the units of velocity and time, 
and therefore indirectly from the units of length and 
time. 

4. In these and all other cases, the practical advantage 
of employing derived units is, that we thus avoid the intro- 
duction of additional factors, which would involve needless 
labour in calculating and difficulty in remembering. * 

5. The correlative term to derived is fundamental. 
Thus, when the units of area, volume, velocity, and 

* An example of such needless factors may be found in the rules 
commonly given in English books for finding the mass of a body 
when its volume and material are given. ' ' Multiply the volume 
in cubic feet by the specific gravity and by 62 '4, and the product 
will be the mass in pounds ; " or " multiply the volume in cubic 



i.] GENERAL THEORY OF UNITS. 7 

acceleration are defined as above, the units of length and 
time are called the fundamental units. 

Dimensions. 

6. Let us now examine the laws according to which 
derived units vary when the fundamental units are 
changed. 

Let V denote a concrete velocity such that a concrete 
length L is described in a concrete time T ; and let v, I, t 
denote respectively the unit velocity, the unit length, and 
the unit time. 

The numerical value of V is to be equal to the numerical 
value of L divided by the numerical value of T. But 

V L T 

these numerical values are , , : 

v L t 

hence we must have 

Hi- o 

This equation shows that, when the units are changed 
(a change which does not affect V, L, and T), v must 
vary directly as I and inversely as t ; that is to say, the 
unit of velocity varies directly as the unit of length, and 

inversely as the unit of time. 

V 
Equation (1) also shows that the numerical value of 

a given velocity varies inversely as the unit of length, and 
directly as the unit of time. 

inches by the specific gravity and by 253, and the product will 
be the mass in grains." The factors 62 '4 and 253 here employed 
would be avoided that is, would be replaced by unity, if the 
unit volume of water were made the unit of mass. 



8 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

7. Again, let A denote a concrete acceleration such 
that the velocity Y is gained in the time T", and let a 
denote the unit of acceleration. Then, since the numerical 
value of the acceleration A is the numerical value of the 
velocity Y divided by the numerical value of the time T', 
we have 



But by equation (1) we may write -for. We 

I T v 

thus obtain 



= 
a I T T' 



This equation shows that when the units , I, t are 
changed (a change which will not affect A, L, T or T'), a 

must vary directly as Z, and inversely in the duplicate 

^ 
ratio of t ; and the numerical value will vary inversely 

a 

as I, and directly in the duplicate ratio of t. In other 
words, the unit of acceleration varies directly as the unit 
of length, and inversely as the square of the unit of time; 
and the numerical value of a given acceleration varies 
inversely as the unit of length, and directly as the square 
of the unit of time. 

It will be observed that these have been deduced as 
direct consequences from the fact that [the numerical 
value of] an acceleration is equal to [the numerical 
value of] a length, divided by [the numerical value 
of] a time, and then again by [the numerical value of] 
a time. 

The relations here pointed out are usually expressed by 



i.] GENERAL THEORY OF UNITS. 9 

savins: that the dimensions of acceleration* are ,- , or 

(time)* 

that the dimensions of the unit of acceleration* are 
unit of length 
(unit of time) 2 

8. We have treated these two cases very fully, by way 
of laying a firm foundation for much that is to follow. 
We shall hereafter use an abridged form of reasoning, 
such as the following : 

, .. length 
velocity = . - : 
time 

, ,. velocity length 
acceleration = : - = T - r ^~--. 
time 



Such equations as these may be called dimensional 
equations. Their full interpretation is obvious from what 
precedes. In all such equations, constant numerical factors 
can be discarded, as not affecting dimensions. 

9. As an example of the application of equation (2) we 
shall compare the unit acceleration based on the foot and 
second with the unit acceleration based on the yard and 
minute. 

Let I denote a foot, L a yard, t a second, T a minute, 
T' a minute. Then a will denote the unit acceleration 
based on the foot and second, and A will denote the unit 

* Professor James Thomson ('Brit. Assoc. Report,' 1878, p. 
452) objects to these expressions, and proposes to substitute the 
following : 

"Change-ratio of ut.it of acceleration =^^ggS&" 
This is very clear and satisfactory as a full statement of the 
meaning intended ; but it is necessary to tolerate some abridg- 
ment of it for practical working. 



10 UNITS AND PHYSICAL CONSTANTS. [CHAP, 

acceleration based on the yard and minute. Equation 
(2) becomes 

A_3 /1\ 2 _ 1 ,ov 

^~I X V60J~1200 ; 
that is to say, an acceleration in which a yard per minute 

of velocity is gained per minute, is *- of an acceleration 

1200 

in which a foot per second is gained per second. 
Meaning of "per." 

10. The word per, which we have several times em- 
ployed in the present chapter, denotes division of the 
quantity named before it by the quantity named after it. 
Thus, to compute velocity in feet per second, we must 
divide a number of feet by a number of seconds.* 

If velocity is continuously varying, let x be the number 
of feet described since a given epoch, and t the number 

of seconds elapsed, then - - is what is meant by the 

at 

number of feet per second. The word should never be 
employed in the specification of quantities, except when 
the quantity named before it varies directly as the quantity 
named after it, at least for small variations as, in the 
above instance, the distance described is ultimately pro- 
portional to the time of describing it. 

Extended Sense of the terms " Multiplication " and 
11 Division" 

11. In ordinary multiplication the multiplier is always- 

* It is not correct to speak of interest at the rate of Five Pounds 
per cent. It should be simply Five pzr cent. A rate of five pounds 
in every hundred pounds is not different from a rate of five 
shillings in every hundred shillings. 



i.] GENERAL THEORY OF UNITS. 11 

a mere numerical quantity, and the product is of the same 
nature as the multiplicand. Hence in ordinary division 
either the divisor is a mere numerical quantity and the 
quotient a quantity of the same nature as the dividend ; 
or else the divisor is of the same nature as the dividend, 
and the quotient a mere numerical quantity. 

But in discussing problems relating to units, it is con- 
venient to extend the meanings of the terms "multiplica- 
tion " and " division." A distance divided by a time 
will denote a velocity the velocity with which the given 
distance would be described in the given time. The dis- 
tance can be expressed as a unit distance multiplied by a 
numerical quantity, and varies jointly as these two factors : 
the time can be expressed as a unit timo multiplied by a 
numerical quantity, and is jointly proportional to these two- 
factors. Also, the velocity remains unchanged when the 
time and distance are both changed in the same ratio. 

12. The three quotients 

1 mile 5280 ft. 22ft. 
1 hour' 3600 sec.' 15 sec. 

all denote the same velocity, and are therefore to be 
regarded as equal. In passing from the first to the 
second, we have changed the units in the inverse ratio 
to their numerical multipliers, and have thus left both 
the distance and the time unchanged. In passing from 
the second to the third, we have divided the two numeri- 
cal factors by a common measure, and have thus changed 
'the distance and the time in the same ratio. A change 
in either factor of the numerator will be compensated 
by a proportional change in either factor of the denom- 
inator. 



1 2 UNITS AND PHYSICAL CONSTANTS. [CHAP. 



Further, since the velocity -' - is -- of the velo- 

15 sec. 15 

.. 1 ft. 22 ft. 22 ft. 

-city v . we are entitled to write - - . , 

1 sec. 15 sec. 15 sec. 

thus separating the numerical part of the expression from 
the units part. 

In like manner we may express the result of Art. 9 by 
writing 

yard 1 foot 

(minute) 2 1200 ' (second) 2 

Such equations as these may be called " physical 
equations," inasmuch as they express the equality of 
physical quantities, whereas ordinary equations express 
the equality of mere numerical values. The use of 
physical equations in problems relating to units is to be 
strongly recommended, as affording a natural and easy 
clue to the necessary calculations, and especially as 
obviating the doubt by which the student is often 
embarrassed as to whether he ought to multiply or 
divide. 

13. In the following examples, which illustrate the use 
of physical equations, we shall employ I to denote the 
unit length, m the unit mass, and t the unit time. 

Ex. 1. If the yard be the unit of length, and the 
acceleration of gravity (in which a velocity of 32*2 ft. per 
sec. is gained per sec.) be represented by 2415, find the 
unit of time. 

We have I = yard, and 

32-2 -A = 2415 l - = 2415 ~ 
(sec.) 2 t* t 2 



t' 2 = p t sec.* = 225 sec. 2 , t = 15 sec. 



i.] GENERAL THEORY OF UNITS. 13 

Ex. 2. If the unit time be the second, the unit density 
162 Ibs. per cub. ft., and the unit force * the weight of an 
ounce at a place where g (in foot-second units) is 32 r 
what is the unit length 1 

We have , = sec., .- 



and - 32 . , or ml =32 oz. ft. = 2 Ib. ft. 
sec. 2 sec.- 

Hence by division 

I* - ^ (ft.)*, jr.'. J It - 4 in. 

Ex. 3. If the area of a field of 10 acres be represented 
by 100, and the acceleration of gravity (taken as 32 foot- 
second units) be 58 1, find the unit of time. 

We have 48400 (yd.) 2 -100 l\ whence 1 = 22 yd.; 

and 



, 

(sec.) 2 ' ? 3 t- 

whence tf = ^- sec. 2 = 121 sec. 2 , =11 sec. 

Ex. 4. If 8 ft. per sec. be the unit velocity, and the 
acceleration of gravity (32 foot-second units) the unit 
acceleration, find the units of length and time. 

We have the two equations 

I -, ft. I oo ft- 

= o - , = o^ - , 
t sec. t- sec.- 

whence by division t = \ sec., and substituting this value 
of t in the first equation, we have 4 =8 ft., 1=2 ft. 

Ex. 5. If the unit force be 100 Ibs. weight, the unit 
length 2 ft., and the unit time J sec., find the unit mass, 
the acceleration of gravity being taken as 32 foot-second 
units. 

* For the dimensions of density and force, see Art. 14. 



14 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

We have I = 2 ft., t = J sec., 

100 Ib. 32 J?L- = ^L = ^_2J*r 
sec. 2 tf Jg- sec. 2 

that is 100 x 32 Ib. = 32 m, m = 100 Ib. 

Ex. 6. The number of seconds in the unit of time is 
equal to the number of feet in the unit of length, the unit 
of force is 750 Ibs. weight [g being 32], and a cubic foot 
of the standard substance [substance of unit density] con- 
tains 13500 oz. Find the unit of time. 

Let t = x sec., then l = xft. also let m = ylb. Then 
we have 

ml = y Ib. x ft. _ y Ib. ft. = 75Q x g9 lb. ft. 
tf x 2 sec. 2 x sec. 2 sec. 2 

or = 750 x 32. 

x 

AT m. y Ib. IOCAA Z - 

Also T -I |p- 13500 a3 ; 

whence U - 13500 x I. 

x 6 lo 

Hence by division 

750 x 32 x 16 16 2 16 , 16 



,. 

Ex. 7. When an inch is the unit of length and t the 
unit of time, the measure of a certain acceleration is a ; 
when 5 ft. and 1 min. are the units of length and time 
respectively, the measure of the same acceleration is 10 a. 
Find t. 

Equating the two expressions for the acceleration, we 

, inch , ~ 5 ft. 

have a = 10 a - : - , 

t~ (mm.) 2 



i.] GENERAL THEORY OF UNITS. 15 

,.0 / \o inch (min.) 2 n / >> 

whence t' = (mm.)" = = 6 (sec.)" 

7 50 ft. 600 

t ^6 sec. 

Ex. 8. The numerical value of a certain force is 56 
when the pound is the unit of mass, the foot the unit of 
length, and the second the unit of time ; what will be the 
numerical value of the same force when the hundredweight 
is the unit of mass, the yard the unit of length, and the 
minute the unit of time 1 

Denoting the required value by x we have 



sec." mm.- 

Ib. ft. 



x = 56 

cwt 



ft. /min.V 
: ^d. \ sea / 



= 56 x _L x 1 x 60 2 = 600. 

Dimensions of Mechanical and Geometrical Quantities. 

14. In the following list of dimensions, we employ the 
letters L, M, T as abbreviations for the words Length, 
Mass, Time. The symbol of equality is used to denote 
sameness of dimensions. 

Area = L 2 , Volume - L 3 , Velocity = -, 
Acceleration = , Momentum = -~. 

Density = , density being defined as mass per unit 
volume. 

Force = -, since a force is measured by the momen- 
tum which it generates per unit of time, and is therefore 



16 UNITS AND PHYSICAL CONSTANTS. [CHAP, 

the quotient of momentum by time or since a force is- 
measured by the product of a mass by the acceleration 
generated in this mass. 

Work = , being the product of force and distance. 

Kinetic energy = -=^-, being half the product of mass 

by square of velocity. The constant factor J can be 
omitted, as not affecting dimensions. 

ML 2 

Moment of couple = ^ , being the product of a force 

by a length. 

The dimensions of angle* when measured by arc 

radius 

are zero. The same angle will be denoted by the same 
number, whatever be the unit of length employed. In 

/. . i arc L T ft 

fact we have - = - == L. 

radius L 

The work done by a couple in turning a body through 
any angle, is the product of the couple by the angle. 
The identity of dimensions between work and couple is 
thus verified. 

Angular velocity 

Angular acceleration = . 

Moment of inertia = ML 2 . 

ML 2 

Angular momentum = moment of momentum = , 

* The name radian has been given to the angle whose arc is 
equal to radius. "An angle whose value in circular measure is 
6 " is " an angle of radians." 



i.] GENERAL THEORY OF UNITS. 17 

being the product of moment of inertia by angular velo- 
city, or the product of momentum by length. 

Intensity of pressure, or intensity of stress generally, 

being force per unit of area, is of dimensions - : that 

area 



Intensity of force of attraction at a point, often called 
simply force at a point, being force per unit of attracted 

mass, is of dimensions - - or . It is numerically 
mass T 2 

equal to the acceleration which it generates, and has 
accordingly the dimensions of acceleration. 

The absolute force of a centre of attraction, better called 
the strength of a centre, may be defined as the intensity of 
force at unit distance. If the law of attraction be that 
of inverse squares, the strength will be the product of the 
intensity of force at any distance by the square of this 

L 3 

distance, and its dimensions will be . 

Curvature (of a curve) = , being the angle turned by 
-L 

the tangent per unit distance travelled along the curve. 
Tortuosity = , being the angle turned by the osculat- 

Ju 

ing plane per unit distance travelled along the curve. 

The solid angle or aperture of a conical surface of any 
form is measured by the area cut off by the cone from a 
sphere whose centre is at the vertex of the cone, divided 
by the square of the radius of the sphere. Its dimensions 
are therefore zero ; or a solid angle is a numerical quan- 
tity independent of the fundamental units. 



18 UNITS AND PHYSICAL CONSTANTS. [CHAP. i. 

The specific curvature of a surface at a given point 
(Gauss's measure of curvature) is the solid angle de- 
scribed by a line drawn from a fixed point parallel to the 
normal at a point which travels on the surface round the 
given point, and close to it, divided by the very small 

area thus enclosed. Its dimensions are therefore . 

The mean curvature of a surface at a given point, in 
the theory of Capillarity, is the arithmetical mean of the 
curvatures of any two normal sections normal to each 

other. Its dimensions are therefore . 

Ju 



19 



CHAPTER II. 
CHOICE OF THREE FUNDAMENTAL UNITS. 

15. NEARLY all the quantities with which physical 
science deals can be expressed in terms of three funda- 
mental units ; and the quantities commonly selected to 
serve as the fundamental units are 

a definite length, 

a definite mass, 

a definite interval of time. 

This particular selection is a matter of convenience 
rather than of necessity ; for any three independent units 
are theoretically sufficient. For example, we might em- 
ploy as the fundamental units 

a definite mass, 

a definite amount of energy, 

a definite density. 

16. The following are the most important considera- 
tions which ought to guide the selection of fundamental 
units : 

(1) They should be quantities admitting of very 
accurate comparison with other quantities of the same 
kind. 



20 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

(2) Such comparison should be possible at all times. 
Hence the standards must be permanent that is, not 
liable to alter their magnitude with lapse of time. 

(3) Such comparisons should be possible at all places. 
Hence the standards must not be of such a nature as 
to change their magnitude when carried from place to 
place. 

(4) The comparison should be easy and direct. 
Besides these experimental requirements, it is also 

desirable that the fundamental units be so chosen that 
the definition of the various derived units shall be easy, 
and their dimensions simple. 

17. There is probably no kind of magnitude which so 
completely fulfils the four conditions above stated as a 
standard of mass, consisting of a piece of gold, platinum, 
or some other substance not liable to be affected by 
atmospheric influences. The comparison of such a 
standard with other bodies of approximately equal 
mass is effected by weighing, which is, of all the 
operations of the laboratory, the most exact. Very ac- 
curate copies of the standard can thus be secured; and 
these can be carried from place to place with little risk 
of injury. 

The third of the requirements above specified forbids 
the selection of a force as one of the fundamental 
units. Forces at the same place can be very accurately 
measured by comparison with weights; but as gravity 
varies from place to place, the force of gravity upon a 
piece of metal, or other standard weight, changes its 
magnitude in travelling from one place to another. A 
spring-balance, it is true, gives a direct measure of 



IT.] THREE FUNDAMENTAL UNITS. 21 

force ; but its indications are too rough for purposes of 
accuracy. 

18. Length is an element which can be very accurately 
measured and copied. But every measuring instrument 
is liable to change its length with temperature. It is 
therefore necessary, in denning a length by reference to a 
concrete material standard, such as a bar of metal, to 
state the temperature at which the standard is correct. 
The temperature now usually selected for this purpose is 
that of a mixture of ice and water (0 C.), observation 
having shown that the temperature of such a mixture is 
constant. 

The length of the standard should not exceed the length 
of a convenient measuring-instrument ; for, in comparing 
the standard with a copy, the shifting of the measuring- 
instrument used in the comparison introduces additional 
risk of error. 

In end-standards, the standard length is that of the bar 
as a whole, and the ends are touched by the instrument 
every time that a comparison is made. This process is 
liable to wear away the ends and make the standard false. 
In line-standards, the standard length is the distance be- 
tween two scratches, and the comparison is made by 
optical means. The scratches are usually at the bottom 
of holes sunk halfway through the bar. 

19. Time is also an element which can be measured 
with extreme precision. The direct instruments of mea- 
surement are clocks and chronometers ; but these are 
checked by astronomical observations, and especially by 
transits of stars. The time between two successive tran- 
sits of a star is (very approximately) the time of the 



22 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

earth's rotation on its axis ; and it is upon the uniformity 
of this rotation that the preservation of our standards of 
time depends. 

Necessity for a Common Scale. 

20. The existence of quantitative correlations between 
the various forms of energy, imposes upon men of science 
the duty of bringing all kinds of physical quantity to one 
common scale of comparison. Several such measures 
(called absolute measures) have been published in recent 
years ; and a comparison of them brings very promi- 
nently into notice the great diversity at present existing 
in the selection of particular units of length, mass, and 
time. 

Sometimes the units employed have been the foot, the 
grain, and the second ; sometimes the millimetre, milli- 
gramme, and second ; sometimes the centimetre, gramme, 
and second ; sometimes the centimetre, gramme, and 
minute ; sometimes the metre, tonne, and second ; some- 
times the metre, gramme, and second ; while sometimes a 
mixture of units has been employed ; the area of a plate, 
for example, being expressed in square metres, and its 
thickness in millimetres. 

A diversity of scales may be tolerable, though undesir- 
able, in the specification of such simple matters as length, 
area, volume, and mass when occurring singly ; for the 
reduction of these from one scale to another is generally 
understood. But when the quantities specified involve 
a reference to more than one of the fundamental units,, 
and especially when their dimensions in terms of these 
units are not obvious, but require careful working out, 



ii.] THREE FUNDAMENTAL UNITS. 23 

there is great increase of difficulty and of liability to 
mistake. 

A general agreement as to the particular units of length, 
mass, and time which shall be employed if not in all 
scientific work, at least in all work involving complicated 
references to units is urgently needed ; and almost any 
one of the selections above instanced would be better than 
the present option. 

21. We shall adopt the recommendation of the Units 
Committee of the British Association (see Appendix), 
that all specifications be referred to the Centimetre, the 
Gramme, and the Second. The system of units derived 
from these as the fundamental units is called the C.G.S. 
system; and the units of the system are called the C.G.S. 
units. 

The reason for selecting the centimetre and gramme, 
rather than the metre and gramme, is that, since a 
gramme of water has a volume of approximately 1 cubic 
centimetre, the former selection makes the density of 
water unity; whereas the latter selection would make 
it a million, and the density of a substance would 
be a million times its specific gravity, instead of being 
identical with its specific gravity as in the C.G.S. 
system. 

Even those who may have a preference for some other 
units will nevertheless admit the advantage of having a 
variety of results, from various branches of physics, re- 
duced from their original multiplicity and presented in 
one common scale. 

22. The adoption of one common scale for all quan- 
tities involves the frequent use of very large and very 



24 UNITS AND PHYSICAL CONSTANTS. [CHAP. n. 

small numbers. Such numbers are most conveniently 
written by expressing them as the product of two factors, 
one of which is a power of 10 ; and it is usually advan- 
tageous to effect the resolution in such a way that the 
exponent of the power of 10 shall be the characteristic of 
the logarithm of the number. Thus 3240000000 will 
be written 3-24 x 10 9 , and '00000324 will be written 
3-24 x 10- 6 . 



25 



CHAPTER III. 

MECHANICAL UNITS. 

Value of g. 

23. ACCELERATION is defined as the rate of increase of 
velocity per unit of time. The C.G.S. unit of accelera- 
tion is the acceleration of a body whose velocity increases 
in every second by the C.G.S. unit of velocity namely, 
by a centimetre per second. The apparent acceleration 
of a body falling freely under the action of gravity in 
vacuo is denoted by g. The value of g in C.G.S. units 
at any part of the earth's surface is approximately given 
by the following formula, 

g = 980-6056 - 2-5028 cos 2/X - -000003/i, 
A denoting the latitude, and h the height of the station 
(in centimetres) above sea-level. 

The constants in this formula have been deduced from 
numerous pendulum experiments in different localities, 
the length I of the seconds' pendulum being connected 
with the value of g by the formula g = Tr 2 l. 

Dividing the above equation by ?r 2 we have, for the 
length of the seconds' pendulum, in centimetres, 
I = 99-3562 - -2536 cos 2A - -0000003A. 



26 



UNITS AND PHYSICAL CONSTANTS. [CHAP. 



At sea-level these formulae give the following values for 
the places specified : 





Latitude. 


Value of g. 


Value of I. 


Equator, - 





978-10 


99-103 


Latitude 45, 


45 


980-61 


99-356 


Munich, - 


48 9 


980-88 


99-384 


Paris, 


48 50 


980-94 


99-390 


Greenwich, 


51 29 


981-17 


99-413 


Gottingen, 


51 32 


981-17 


99-414 


Berlin, - 


52 30 


981-25 


99-422 


Dublin, - 


53 21 


981-32 


99-429 


Manchester, 


53 29 


981-34 


99-430 


Belfast, - 


54 36 


981-43 


99-440 


Edinburgh, 


55 57 


981-54 


99-451 


Aberdeen, 


57 9 


981-64 


99-461 


Pole, 


90 


983-11 


99-610 



The difference between the greatest and least values 

(in the case of both g and 1) is about of the mean 

j. y o 

value. 

24. The Standards Department of the Board of Trade, 
being concerned only with relative determinations, has 
adopted the formula 

g = <7 (1- -00257 cos 2A)(l - | A), 

A. denoting the latitude, k the height above sea-level, K 
the earth's radius, g the value of g in latitude 45 at sea- 
level, which may be treated as an unknown constant 
multiplier. Putting for E, its value in centimetres, the 
formula gives 

g = g (l- -00257 cos 2A.-1 -96/i x 10' 9 ), 
where h denotes the height in centimetres. 



in.] MECHANICAL UNITS. 27 

The formula which we employed in the preceding 
section gives 



As regards the factor dependent on height, theory indi- 
cates 1 - as its correct value for such a case as that of 

XV 

a balloon in mid-air over a low-lying country ; the value 

5 h 

1 -- may be accepted as more correct for an elevated 

4: .TV 

plateau on the earth's surface. 

Force. 

25. The C.G.S. unit of force is called the dyne. It is 
the force which, acting upon a gramme for a second, 
generates a velocity of a centimetre per second. 

It may otherwise be denned as the force which, acting 
upon a gramme, produces the C.G.S. unit of acceleration, 
or as the force which, acting upon any mass for 1 second, 
produces the C.G.S. unit of momentum. 

To show the equivalence of these three definitions, let 
m denote mass in grammes, v velocity in centimetres per 
second, t time in seconds, F force in dynes. 

Then, by the second law of motion, we have 

, . . force 
acceleration = - : 
mass 

"IT 

that is, if a denote acceleration in C.G.S. units, a=- ; 

m 

hence, when a and m are each unity, F will be unity. 

Again, by the nature of uniform acceleration, we have 
v = at, v denoting the velocity due to the acceleration a, 
continuing for time t. 



28 UNITS AND PHYSICAL CONSTANTS. [CHAP. 



Hence we have F = ma = . Therefore, if mv = 1 

and t=l t we have F = 1. 

As a particular case, if w = l, vl, t = l, we have 



26. The force represented by the weight of a gramme 
varies from place to place. It is the force required to 
sustain a gramme in vacuo, and would be nil at the 
earth's centre, where gravity is nil. To compute its 
amount in dynes at any place where g is known, observe 
that a mass of 1 gramme falls in vacuo with acceleration 
g. The force producing this acceleration (namely, the 
weight of the gramme) must be equal to the product of 
the mass and acceleration, that is, to g. 

The weight (when weight means force) of 1 gramme is 
therefore g dynes ; and the weight of in grammes is mg 
dynes. 

27. Force is said to be expressed in gravitation-measure 
when it is expressed as equal to the weight of a given 
mass. Such specification is inexact unless the value of 
g is also given. For purposes of accuracy it must always 
be remembered that the pound, the gramme, etc., are, 
strictly speaking, units of mass. Such an expression as 
" a force of 100 tons " must be understood as an abbrevia- 
tion for " a force equal to the weight [at the locality in 
question] of 100 tons." 

28. The name poundal has recently been given to the 
unit force based on the pound, foot, and second ; that is, 
the force which, acting on a pound for a second, gene- 

rates a velocity of a foot per second. It is of the 



in.] MECHANICAL UNITS. 29 

weight of a pound, y denoting the acceleration due to 
gravity expressed in foot-second units, which is about 
32-2 in Great Britain. 

To compare the poundal with the dyne, let x denote 
the number of dynes in a poundal ; then we have 
mn. cm. Ib. ft. 



x 1 - 



sec." 1 sec.- 



x = . = 453-59 x 30-4797 = 13825. 
gm. cin. 



Work and Energy. 

29. The C.G.S. unit of work is called the erg. It is 
the amount of work done by a dyne working through a 
distance of a centimetre. 

The C.G.S. unit of energy is also the erg, energy being 
measured by the amount of work which it represents. 

30. To establish a rule for computing the kinetic energy 
(or energy due to the motion) of a given mass moving with 
a given velocity, it is sufficient to consider the case of 
a body falling in vacuo. 

When a body of m grammes falls through a height of h 
centimetres, the working force is the weight of the body 
that is, gm dynes, which, multiplied by the distance 
worked through, gives gmh ergs as the work done. But 
the velocity acquired is such that v~ = 2gh. Hence we 
have gmh = ^mv 2 . 

The kinetic energy of a mass of m grammes moving 
with a velocity of v centimetres per second is therefore 
^mv 2 ergs ; that is to say, this is the amount of work 
which would be required to generate the motion of 
the body, or is the amount of work which the body 



30 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

would do against opposing forces before it would coine 
to rest. 

31. Work, like force, is often expressed in gravitation- 
measure. Gravitation units of work, such as the foot- 
pound and kilogramme-metre, vary with locality, being 
proportional to the value of g. 

One gramme-centimetre is equal to g ergs. 
One kilogramme-metre is equal to 100,000 g ergs. 
One foot-poundal is 453'59 x (30'4797) 2 = 421390 ergs. 
One foot-pound is 13,825 g ergs, which, if g be taken 
as 981, is 1-356 x 10 7 ergs. 

32. The C.G.S. unit rate of working is 1 erg per second. 
Watt's " horse-power " is denned as 550 foot-pounds per 
second. This is 7 '46 x 10 9 ergs per second. The " force de 
cheval " is denned as 75 kilogrammetres per second. This 
is 7 '36 x 10 9 ergs per second. We here assume g 981. 

A new unit of rate of working has been lately intro- 
duced for convenience in certain electrical calculations. 
It is called the Watt, and is denned as 10 7 ergs per second. 
A thousand watts make a kilowatt. The following 
tabular statement will be useful for reference. 

1 Watt = 10 7 ergs per second = '00134 horse-power 
= 737 foot-pounds per second = '101 9 kilogram- 
metres per second. 
1 Kilowatt = 1'34 horse-power. 

1 Horse-power = 550 foot-pounds per second = 76'0 
kilogrammetres per second = 746 watts = 1 '01385 
force de cheval. 

1 Force de cheval = 75 kilogrammetres per second 
= 542 '48 foot-pounds per second = 736 watts 
= '9863 horse-power. 



m.l MECHANICAL UNITS. 31 

Examples. 

1. If a spring balance is graduated so as to show the 
masses of bodies in pounds or grammes when used at the 
equator, what will be its error when used at the poles, 
neglecting effects of temperature 1 

Ans. Its indications will be too high by about j-gg of 

the total weight. 

2. A cannon-ball, of 10,000 grammes, is discharged 
with a velocity of 45,000 centims. per second. Find its 
kinetic energy. 

Ans. | x 10000 x (45000) 2 = 1-0125 x 10 13 ergs. 

3. In last question find the mean force exerted upon 
the ball by the powder, the length of the barrel being 
200 centims. 

Ans. 5-0625 x 10 10 dynes. 

4. Given that 42 million ergs are equivalent to 1 
gramme-degree of heat, and that a gramme of lead at 
10 C. requires 15 '6 gramme-degrees of heat to melt it; 
find the velocity with which a leaden bullet must strike a 
target that it may just be melted by the collision, suppos- 
ing all the mechanical energy of the motion to be converted 
into heat and to be taken up by the bullet. 

We have Jv 2 = 15-6 x J, where J = 42 x 10 6 . Hence 
i> 2 =1310 millions; -v-36'2 thousand centims. per 
second. 

5. With what velocity must a stone be thrown verti- 
cally upwards at a place where gis 981 that it may rise 
to a height of 3000 centims. ? and to what height would 
it ascend if projected vertically with this velocity at the 
surface of the moon, where g is 150 1 

Ans. 2426 centims. per second ; 19620 centims. 



32 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

Centrifugal Force. 

33. A body moving in a curve must be regarded as 
continually falling away from a tangent. The accelera- 
tion with which it falls away is , v denoting its velocity 

and r the radius of curvature. The acceleration of a 
body in any direction is always due to force urging it in 
that direction, this force being equal to the product of 
mass and acceleration. Hence the normal force on a body 
of in grammes moving in a curve of radius r centimetres, 

with velocity v centimetres per second, is dynes. This 

force is directed towards the centre of curvature. The 
equal and opposite force with which the body reacts is 
called centrifugal force. 

If the body moves uniformly in a circle, the time 

of revolution being T seconds, we have v = ^ ', 



hence ; == (^p~) r > an d * ne force acting on the body is 



nes. 



If n revolutions are made per minute, the value of T is 

60 (mr\- , 

, and the force is mr\ ^1 dynes. 



Examples. 

1. A body of m grammes moves uniformly in a circle 
of radius 80 centims., the time of revolution being J of a 



in.] MECHANICAL UNITS. 33 

second. Find the centrifugal force, and compare it with 
the weight of the body. 

Ans. The centrifugal force ismx/-^j x80 = mx 647r 2 

x 80 = 50532 m dynes. 

The weight of the body (at a place where g is 981) is 
981 m- dynes. Hence the centrifugal force is about 52| 
times the weight of the body. 

.2. At a bend in a river, the velocity in a certain part 
of the surface is 170 centims. per second, and the radius 
of curvature of the lines of flow is 9100 centims. Find 
the slope of the surface in a section transverse to the lines 
of flow. 

Ans. Here the centrifugal force for a gramme of the 
water is ( 17 ^ 8 = 3-176 dynes. If g be 98 1 the slope will 



o .1 fr f* 1 

be = ; that is, the surface will slope upwards 

i/ol oUJ 

from the concave side at a gradient of 1 in 309. The 
general rule applicable to questions of this kind is that 
the resultant of centrifugal force and gravity must be 
normal to the surface. 

3. An open vessel of liquid is made to rotate rapidly 
round a vertical axis. Find the number of revolutions 
that must be made per minute in order to obtain a slope 
of 30 at a part of the surface distant 10 centims. from 
the axis, the value of g being 981. 

Ans. We must have tan 30= /, where /denotes the 

9 

intensity of centrifugal force that is, the centrifugal force 

per unit mass. We have therefore 

c 



34 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

981 tan 30 = lof Y' n denotin the number of 
\30/ revolutions per minute, 

w V 
~~90~* 
Hence rc = 71'9. 

4. For the intensity of centrifugal force at the equator 
due to the earth's rotation, we have r = earth's radius 
= 6-38 x 10 8 , T = 86164, being the number of seconds in 
a sidereal day. 



This is about of the value of g. 

If the earth were at rest, the value of g at the equator 
would be greater than at present by this amount. If the 
earth were revolving about 17 times as fast as at present, 
the value of g at the equator would be nil. 

SUPPLEMENTAL SECTION. 

On the help to be derived from Dimensions in investi- 
gating Physical Formulae. 
-. 
When one physical quantity is known to vary as some 

power of another physical quantity, it is often possible to 
find the exponent of this power by reasoning based on 
dimensions, and thus to anticipate the results or some 
of the results of a dynamical investigation. 

Examples. 

1. The time of vibration of a simple pendulum in a 
small arc depends on the length of the pendulum and the 
intensity of gravity. If we assume it to vary as the m th 



in.] MECHANICAL UNITS. 35 

power of the length, and as the n th power of #, and to be 
independent of everything else, the dimensions of a time 
must equal the m th power of a length, multiplied by the n th 
power of an acceleration, that is 

T = LLT- 2M = L w L M T- 2n 



Since the dimensions of both members are to be identical, 
we have, by equating the exponents of T, 
1 = - 2n, whence n = - J, 
and by equating the exponents of L, 

m + n = 0, whence m = J ; 

that is, the time of vibration varies directly as the square 
root of the length, and inversely as the square root of g. 

2. The velocity of sound in a gas depends only on the 
density D of the gas and its coefficient of elasticity E, and 
we shall assume it to vary as D m E". 

The dimensions of velocity are LT~ T . 

The dimensions of density, or - - , are ML~ 3 . 

volume 

The dimensions of E, which will be explained in the 

chapter on stress and strain, are - , or (MLT' 2 )L~ 2 , or 

area 

ML- 1 T- 2 . 

The equation of dimensions is 

LT _i = MW L _ 3m MW L _ n T _ 2M) 

= M m+ " L~ 3w - n T~ 2w , 

whence, by equating coefficients, we have the three 
equations 

1 = 3m ri) 1 = 2n, m + n = 0, 
to determine the two unknowns m and n. 



36 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

The second equation gives at once 

n = J. 

The third then gives 

m = - 1, 

and these values will be found to satisfy the first equation 
also. 

The velocity, then, varies directly as the square root of 
E, and inversely as the square root of D. 

3. The frequency of vibration f for a musical string 
(that is, the number of vibrations per unit time) depends 
on its length I, its mass m, and the force with which it 
is stretched F. 

The dimensions of f are T" 1 . 

F MLT- 2 . 

Assume that f varies as l x m y ^ z . Then we have 



giving - 1 = - 
whence = 



VTT 
. 



4. The angular acceleration of a uniform disc round its 
axis depends on the applied couple G, the mass of the disc 
M, and its radius R. 

Assume it to vary as G x M. y E, 2 . 

The dimensions of angular acceleration are T~' 2 . 

G ML 2 T- 2 . 

R I*. 

Hence we have 



m.] MECHANICAL UNITS. 37 

giving - 2 = - 2x, x + y - 0, 2x + z - 0, 
whence x-1, ?/=], z-2. 

Hence the angular acceleration varies as 2 . 

JVL.LV 

In the following example the information obtained is 
less complete : 

5. The range of a projectile on a horizontal plane 
through the point of projection depends on the initial 
velocity V, the intensity of gravity g, and the angle of 
elevation a. 

The dimensions of range are L. 

V T/T- 1 

)) 5) 3) AJ - L 

5) 5) 9 ?) * J * " 

,, a, LT, and the dimensions 

of all powers of a are LT. Hence we can draw no 
inferences as to the manner in which a enters the expres- 
sion for the range. The dimensions of this expression will 
depend upon Y and g alone. 

Assume that the range varies as V m g n . Then 

T m-\-n HP m In . 



giving 

whence m-2, n - 1. 

"V 2 
Hence the range varies as when a is given. 

J 



38 



CHAPTER IV, 
HYDROSTATICS. 

34. THE following table of the relative density of water 
at various temperatures (under atmospheric pressure), the 
density at 4 C. being taken as unity, is from Rossetti's 
results deduced from all the best experiments (Ann. Ch. 
Phys. x. 461 ; xvii. 370, 1869) :- 



Temp. 
Cent. 


Relative 
Density. 


Temp. 
Cent. 


Relative 
Density. 


Temp. 

Cent. 


Relative 
Density. 







999871 


13 


999430 


35 


99418 


1 


999923 


14 


999299 


40 


99235 


2 


999969 


15 


999160 


45 


99037 


3 


999991 


16 


999002 


50 


98820 


4 


1-000000 


17 


998841 


55 


98582 


5 


999990 


18 


998654 


60 


98338 


6 


999970 


19 


998460 


65 


98074 


7 


999933 


20 


998259 


70 


97794 


8 


999886 


22 


997826 


75 


97498 


9 


999824 


24 


997367 


80 


97194 


10 


999747 


26 


996866 


85 


96879 


11 


999655 


28 


996331 


90 


96556 


12 


999549 


30 


995765 


100 


95865 



35. According to Kupffer's observations, as reduced 
by Professor W. H. Miller, the absolute density (in 
grammes per cubic centimetre) at 4 is not 1, but 
1-000013. Multiplying the above numbers by this 



CHAP. IV.] 



HYDROSTATICS. 



39 



factor, we obtain the following table of absolute den- 
sities : 



Temp 


Density. 


Temp. 


Density. | Temp. 


Density. 


6 


999884 


13 


999443 


35 


99469 


i 


999941 


14 


999312 


40 


99236 


2 


999982 


15 


999173 


45 


99038 


3 


1-000004 


16 


999015 


50 


98821 


4 


1-000013 


17 


998854 


55 


98583 


5 


1-000003 


18 


998667 60 


98339 


6 


999983 


19 


998473 


65 


98075 


7 


999946 


20 


998272 


70 


97795 


8 


999899 


22 


997839 


75 


97499 


9 


999837 


24 


997380 


80 


97195 


10 


999760 


26 


996879 


85 


96880 


11 


999668 


28 


996344 


90 


96557 


12 


999562 


30 


995778 


100 


95866 



36. The volume, at temperature t\ of the water which 
occupies unit volume at 4, is approximately 

1 + A(*-4) 2 -B(-4) 2 - 6 + C(*-4) 3 , 
where 

A = 8-38 x 10- 6 , 
B = 3-79 x 10- 7 , 
C = 2-24 x 10- 8 ; 

and the relative density at temperature t is given by the 
same formula with the signs of A, B, and C reversed. 
The rate of expansion at temperature t is 



In determining the signs of the terms with the frac- 
tional exponents 2*6 and 1*6, these exponents are to be 
regarded as odd. 

37. The following Table of Densities has been compiled 
by collating the best authorities, but is only to be taken 



40 



UNITS AND PHYSICAL CONSTANTS. [CHAP. 



as giving rough approximations. Most of the densities 
vary between wide limits in different specimens : 

Solids. 



Aluminium, 
Antimony, 


2-6 
6-7 


Carbon (diamond),., 
(graphite),.. 
,, (gas carbon), 
,, (wood charcoal), 
Phosphorus (ordi- 
nary), 
(red),... 
Sulphur (roll), 
Quartz (rock cry- 
stal), 
Sand (dry), 
Clay, . 


3-5 
2-3 
1-9 
1-6 

1-83 
2-2 
2-0 

2-65 
1-42 
1-9 
2-1 
3-0 
8 to 2-8 
5 to 2-7 
to 3-5 
2-4 


Bismuth 


9'8 


Brass, 


8-4 


Copper, 
Gold, 


8-9 
19-3 


Iron, 


7'8 


Lead, . . . 


11 -8 


Nickel, 
Platinum, 


8-9 
21-5 


Silver, 


10-5 


Sodium, . . 


98 


Tin 


7'3 


Brick, 
Basalt 


Zinc 


7'1 


Cork, 


24 


Chalk 1 


Oak, .. 


. 7 to 1 '0 


Glass (crown), 2 
,, (flint), 3 


Ebony, 
Ice. .'. 


l-ltol-2 
918 


Porcelain . . . 



1 -026 
8 
1-5 
73 


Sulphuric Acid,... . 
Nitric Acid, 
Hydrochloric Acid, 
Milk, 


1-85 
1-56 
1-27 
1-03 


1-29 
1-27 
3'5'JG 


Oil of Turpentine,.. 
,, Linseed, 
,, Mineral, 


87 
94 
76 to -83 



Liquids at C. 

Sea water, 

Alcohol, 

Chloroform, 

Ether, 

Bisulphide of Carbon,.. 

Glycerine, T27 

Mercury, 13'5 ( J6 

More exactly, the density of mercury at C., as com- 
pared with water at the temperature of maximum density, 
under atmospheric pressure, is 13*5956. 

38. If a body weighs m grammes in vacuo and ra' 
grammes in water of density unity, the volume of the 
body is m - m' cubic centims. ; for the mass of the water 
displaced is m - m' grammes, and each gramme of this 
water occupies a cubic centimetre. 



TV.] HYDROSTATICS. 41 

Examples. 

1. A glass cylinder, I centims. long, weighs m grammes 
in vacuo and m' grammes in water of unit density. Find 
its radius. 

Solution. Its section is Trr 2 , and is also m ~ m ; hence 

L 

o m m! 

r 2 = 

Trl 

2. Find the capacity at C. of a bulb which holds m 
grammes of mercury at that temperature. 

Solution. The specific gravity of mercury at being 
13 '5 95 6 as compared with water at the temperature of 
maximum density, it follows that the mass of 1 cubic 
centirn. of mercury is 13-5956 x 1-000013 = 13-5958, say 

13 '5 9 6. Hence the required capacity is cubic 

lo '0\y O 

centims. 

3. Find the total pressure on a surface whose area is A 
square centims. when its centre of gravity is immersed to 
a depth of h centims. in water of unity density, atmos- 
pheric pressure being neglected. 

Ans. A.h grammes weight ; that is, gAk dynes. 

4. If mercury of specific gravity 13-596 is substituted 
for water in the preceding question, find the pressure. 

Ans. 13-596 AJi grammes weight ; that is, 13*596 gAJi 
dynes. 

5. If h be 76, and A be unity in example 4, the answer 
becomes 1033*3 grammes weight, or 1033 -3g dynes. 

For Paris, where g is 980-94, this is I'0136xl0 6 
dynes. 



42 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

Harometric Pressure. 

39. The C.G.S. unit of pressure intensity (that is, of 
pressure per unit area) is the pressure of a dyne per 
square centim. 

At the depth of h centiins. in a uniform liquid whose 
density is d [grammes per cubic centim.], the pressure due 
to the weight of the liquid is ghd dynes per square centim. 

The pressure-intensity due to the weight of a column of 
mercury at C., 76 centhns. high, is found by putting 
&=76, d= 13*596, and is 1033%. It is therefore 
different at different localities. At Paris, where g is 
980-94, it is 1-0136 x 10 6 ; that is, rather more than a 
megadyne * per square centim. To exert a pressure of 
exactly one megadyne per square centim., the height of 
the column at Paris must be 74*98 centims. 

At Greenwich, where g is 981-17, the pressure due to 
76 centims. of mercury at C. is 1*0138 x 10 6 ; and the 
height which would give a pressure of 10 6 is 74*964 
centims., or 29-514 inches. 

Convenience of calculation would be promoted by 
adopting the pressure of a megadyne per square centim., 
or 10 6 C.G.S. units of pressure-intensity, as the standard 
atmosphere. 

The standard now commonly adopted (whether 76 
centims. or 30 inches) denotes different pressures at 
different places, the pressure denoted by it being pro- 
portional to the value of g. 

We shall adopt the megadyne per square centim. as 
our standard atmosphere in the present work. 

*The prefix mega denotes multiplication by a million. A 
megadyne is a force of a million dynes. 



iv.] HYDROSTATICS. 4 a 

Examples. 

1. What must be the height of a column of water of 
unit density to exert a pressure of a megadyne per square 
centim. at a place where g is 981 1 

Ans. 1000Q00 - 1019-4 centims. This is 33-445 feet. 

i/o JL 

2. What is the pressure due to an inch of mercury at 
C. at a place where g is 981 1 (An inch is 2*54 
centims.) 

Ans. 981 x 2-54 x 13*596 = 33878 dynes per square 
centim. 

3. What is the pressure due to a centim. of mercury at 
C. at the same locality ? 

Ans. 981 x 13-596 = 13338. 

4. What is the pressure due to a kilometre of sea- water 
of density 1-027, g being 981 1 

Ans. 981 x 10 5 x 1 -027 = 1'0075 x 10 s dynes per square 
centim., or 1-0075 x 10 2 megadynes per square centim. ; 
that is, about 100 atmospheres. 

5. What is the pressure due to a mile of the same 
water 1 

Ans. 1-6214 x 10 8 C.G.S. units, or 162-14 atmospheres 
[of a megadyne per square centim.]. 

Density of Air. 

40. Regnault found that at Paris, under the pressure 
of a column of mercury at 0, of the height of 76 centims., 
the density of perfectly dry air was '0012932 gramme per 
cubic centim. The pressure corresponding to this height 
of the barometer at Paris is 1 -0136 x 10 6 dynes per square 



44 UNITS AND PHYSICAL CONSTANTS. [CHAP- 

centini. Hence, by Boyle's law, we can compute the 
density of dry air at 0. at any given pressure. 

At a pressure of a megadyne (10 6 dynes) per square 



centini. the density will be . . " = -0012759. 

1 "Olob 

The density of dry air at C. at any pressure p (dynes 
per square centim.) is 

PY. l-2759x!0- 9 .... (4) 

Example. 

Find the density of dry air at C., at Edinburgh, 
under the pressure of a column of mercury at C., of 
the height of 76 centims. 

Here we have p = 981 -54 x 76 x 13-596 = 1-0142 x 10 6 . 
Ans, Required density = 1-2940 x 10~ 3 = -0012940 
gramme per cubic centim. 

41. Absolute Densities of Gases, in grammes per cubic 
centim., at C., and a pressure of 10 6 dynes per 
square centim. 

Mass of a cubic Volume of a gramme 
centim. in grammes, in cubic centims. 

Air, dry, .................. -0012759 ......... 783"S 

Oxygen, .................... "0014107 ......... 70S"9 

Nitrogen, ................... -0012393 ......... 806"9 

Hydrogen, ................. "00008837 ........ 11316-0 

Carbonic Acid, ............ "0019509 ......... 512-6 

Oxide, .......... -0012179 ......... 821-1 

Marsh Gas, ................ "0007173 ......... 1394-1 

Chlorine, .................... "0030909 ......... 323'5 

Protoxide of Nitrogen,.. "0019433 ......... 514-6 

Binoxide ,, ... "0013254 ......... 754-5 

Sulphurous Acid, ........ '0026990 ......... 370-5 

Cyanogen, .................. "0022990 ......... 435-0 

OlefiantGas, ............... -0012529 ......... 798-1 

Ammonia,.. "0007594 1316 "8 



iv.] HYDROSTATICS. 45 

The numbers in the second column are the reciprocals 
of those in the first. 

The numbers in the first column are identical with the 
specific gravities referred to water as unity. 

Assuming that the densities of gases at given pressure 
and temperature are directly as their atomic weights, we 
have for any gas at zero 

pvp= 1-1316 xl0 10 m; 

v denoting its volume in cubic centims., m its mass in 
grammes, p its pressure in dynes per square centim., and 
/A its atomic weight referred to that of hydrogen as unity. 

Height of Homogeneous Atmosphere. 

42. We have seen that the intensity of pressure at 
depth 7i, in a fluid of uniform density d, is ghd when the 
pressure at the upper surface of the fluid is zero. 

The atmosphere is not a fluid of uniform density ; but 
it is often convenient to have a name to denote a height 
H such that p = #HD, where p denotes the pressure and 
D the density of the air at a given point. 

It may be defined as the height of a column of uniform 
fluid having the same density as the air at the point, 
which would exert a pressure equal to that existing at 
the point. 

If the pressure be equal to that exerted by a column of 
mercury of density 13'596 and height h, we have 
p=ghx 13-596; 

.-. HD = A x 13-596, H = Ax18 ' 596 . 

If it were possible for the whole body of air above the 
point to be reduced by vertical compression to the density 
which the air has at the point, the height from the point 



46 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

up to the summit of this compressed atmosphere would be 
equal to H, subject to a small correction for the variation 
of gravity with height. 

H is called the height of the homogeneous atmosphere at 
the point considered. Pressure-height would be a better 
name. 

The general formula for it is 

H-; ... (5) 

and this formula will be applicable to any other gas as 
well as dry air, if we make D denote the density of the 
gas (in grammes per cubic centim.) at pressure p. 

If, instead of p being given directly in dynes per square 
centim., we have given the height h of a column of liquid 
of density d which would exert an equal pressure, the 
formula reduces to 

H = j (6) 

43. The value of in formula (5) depends only on the 

nature of the gas and on the temperature ; hence, for a 
given gas at a given temperature, H varies inversely 
as g only. 

For dry air at zero we have, by formula (4), 



7-8376 x 10 8 
1 = - . 

9 
At Paris, where g is 9 80 '9 4, we find 

H = 7-990 xlO 5 . 
At Greenwich, where g is 981-17, 

H = 7-988 xlO 5 . 



iv.] HYDROSTATICS. 47 

Examples. 

1. Find the height of the homogeneous atmosphere at 
Paris for dry air at 10 C., and also at 100 C. 

Ans. For given density, p varies as 1 x -00366 t, t de- 
noting the temperature on the Centigrade scale. Hence 
we have, at 10 C., 

H - 1-0366 x 7-99 x 10 5 = 8-2825 x 10 5 ; 
and at 100 C., 

H = 1-366 x 7-99 x 10 5 - 1-0914 x 10 6 . 

2. Find the height of the homogeneous atmosphere for 
hydrogen at 0, at a place where g is 981. 

Here we have 



Diminution of Density with increase, of Height in the 
Atmosphere. 

44. Neglecting the variation of gravity with height, 
the variation of H as we ascend in the atmosphere would 
depend only on variation of temperature. In an atmos- 
phere of uniform temperature H will be the same at all 
heights. In such an atmosphere, an ascent of 1 centim. 
will involve a diminution of the pressure (and therefore 

of the density) by of itself, since the layer of air which 

has been traversed is -- of the whole mass of superincum- 
H 

bent air. The density therefore diminishes by the same 
fraction of itself for every centim. that we ascend; in 
other words, the density and pressure diminish in geo- 
metrical progression as the height increases in arithmetical 
progression. 



48 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

Denote height above a fixed level by x, and pressure 
by p. Then, in the notation of the differential calculus, 

we have $=-^, 

II p 

and if p lt p 2 are the pressures at the heights x lt x z , we 
deduce 

x 2 - a?! - H log.Pl = H x 2-3026 log TO &. . . (7) 

Pz Pz 

In the barometric determination of heights it is usual 
to compute H by assuming a temperature which is the 
arithmetical mean of the temperatures at the two heights. 
For the latitude of Greenwich formula (7) becomes 

a? 2 - a^ = (1 x -00366 t) 7'988 x 10 5 x 2*3026 log^l 

P 2 
= (1 x -00366 1) 1,839,300 log^l, . . (8) 

P2 

t denoting the mean temperature, and the logarithms 
being common logarithms. 

To find the height at which the density would be halved, 
variations of temperature being neglected, we must put 2 

for O in these formulae. The required height will be H 

Pi 

log e 2, or, in the latitude of Greenwich, for temperature 
0., will be 

1-8393 x 10 6 x -30103 - 553700. 
The value of log e 2, or 2-3026 Iog 10 2, is 
2-3026 x -30103- -69315. 

Hence for an atmosphere of any gas at uniform tempera- 
ture, the height at which the density would be halved is 
the height of the homogeneous atmosphere for that gas, 
multiplied by '69315. The gas is assumed to obey 
Boyle's law. 



HYDROSTATICS. 



49 



Examples. 

1. Show that if the pressure of the gas at the lower 
station and the value of g be given, the height at which 
the density will be halved varies inversely as the density. 

2. At what height, in an atmosphere of hydrogen at 
C., would the density be halved, g being 98 1 1 ? 

Ana. 7-9954 x 10 6 . 

45. The phenomena of capillarity, soap-bubbles, etc., 
can be reduced to quantitative expression by assuming a 
tendency in the surface of every liquid to contract. The 
following table exhibits the intensity of this contractile 
force for various liquids at the temperature of 20 C. 
The contractile force diminishes as the temperature in- 
creases. 

Superficial tensions at 20 C., in dynes per linear centim., 
deduced from Quincke's results. 







Tension of Surface separating 
the Liquid from 




Density. 








Air. 


Water. 


Mercury. 


Water, 


0-9982 


81 





418 


Mercury, 


13-5432 


540 


418 





Bisulphide of Carbon, 


1-2687 


32-1 


41-75 


372-5 


Chloroform, 


1-4878 


30-6 


29-5 


399 


Alcohol, 


7906 


25-5 




399 


Olive Oil, 


9136 


36-9 


20 : 56 


335 


Turpentine, 


8867 


29-7 


11-55 


250-5 


Petroleum, 


7977 


31-7 


27'8 


284 


Hydrochloric Acid, 


1-1 


70-1 




377 


Solution of Hyposul- ) 
phite of Soda, - - ) 


1-1248 


77-5 




442-5 



The values here given for water and mercury are only 
applicable when special precautions are taken to ensure 



50 



UNITS AND PHYSICAL CONSTANTS. [CHAP. 



cleanliness and purity. Without such precautions smaller 
values will be obtained. (Quincke in Wied. Ann., 1886, 
page 219.) 

The following values are from the observations of A. M. 
Worthington (Proc. Roy. Soc., June 16, 1881), at tempera- 
tures from 15 to 18 C., for surfaces exposed to air : 

Surface Tension. 

In gm. per cm. In dynes per cm. 

Water, '072 to '080 70'6 to 78'5 

Alcohol, -02586 25'3 

Turpentine, '02818 27 '6 

Olive Oil, -03373 33'1 

Chloroform, '03025 29 "6 

46. Very elaborate measurements of the thicknesses of 
soap films have been made by Reinold and Riicker (Phil. 
Trans., 1881, p. 456 ; and 1883, p. 651). When so thin 
as to appear black, the thickness varied from 7 "2 to 14 -5 
millionths of a millimetre, the mean being 11 '7. This is 
1*17 x 10~ 6 centimetre. The following thicknesses were 
observed for the colours of the successive orders : 



Thickness. 
cm. 

FIRST ORDER 

Red, 2'84xlO- 6 

SECOND ORDER 

Violet, 3'05 

Blue, 3-53 

Green, 4'09 

Yellow, 4-54 

Orange, 4'91 

Red, 5'22 

THIRD ORDER 

Purple, 5'59 

Blue, 5'77 

,, 6-03 

Green, 6'56 



Thickness. 
cm. 

Yellow, 7.10xlO- 5 

Red, 7-65 

Bluish Red, 8'15 ,, 

FOURTH ORDER 

Green, 8'41 ,, 

8 '93 , , 

Yellow-Green,.. 9 '64 ,, 

Red, 10-52 

FIFTH ORDER 

Green, M19xlO- 4 

1-188 , 



Red,, 



1-260 
1-335 



iv.] 



HYDROSTATICS. 



51 



Thickness, 
cm. 

SIXTH ORDER 

Green, l'410x!0- 4 

1-479 

Red, 1-548 

1-627 ! 



SEVENTH ORDER 
Green,.... 1 



705 



Thickness, 
cm. 

Green, 1787 xlO~ 4 

Red, 1-869 

1-936 

EIGHTH ORDER 

Green, 2'004 ,, 

Red,.. , 2-115 , 



4 6 A. Depression of the barometrical column due to 
capillarity, according to Pouillet : 



Internal 
Diameter 
of tube. 



2 

2-5 

3 

3-5 

4 

4-5 

5 

5-5 

6 

6-5 

7 

7'5 

8 





Internal 


Depression. 


Diameter 




of tube. 


mm. 


mm. 


4-579 


8-5 


3-595 


9 


2-902 


9-5 


2-415 


10 


2-053 


10-5 


1-752 


11 


1-507 


11-5 


1-306 


12 


1-136 


12-5 


995 


13 


877 


13-5 


775 


14 


684 


14-5 





Internal 


Depression. 


Diameter 




of tube. 


mm. 


mm. 


604 


15 


534 


15-5 


473 


16 


419 


16-5 


372 


17 


330 


17-5 


293 


18 


750 


18-5 


230 


19 


204 


19-5 


181 


20 


161 


20-5 


143 


21 



Depression, 
mm. 

127 
112 
099 
087 
077 
068 
060 
053 
047 
041 
036 
032 
028 



52 



CHAPTER V. 
STRESS, STRAIN, AND RESILIENCE. 

47. IN the nomenclature introduced by Rankine, and 
adopted by Thomson and Tait, any change in the shape 
or size of a body is called a strain, and an action of force 
tending to produce a strain is called a stress. We shall 
always suppose strains to be small ; that is, we shall sup- 
pose the ratio of the initial to the final length of every 
line in the strained body to be nearly a ratio of equality. 

48. A strain changes every small spherical portion of 
che body into an ellipsoid ; and the strain is said to be 
homogeneous when equal spherical portions in all parts 
of the body are changed into equal ellipsoids with their 
corresponding axes equal and parallel. When the strain 
consists in change of volume, unaccompanied by change 
of shape, the ellipsoids are spheres. 

When strain is not homogeneous, but varies continu- 
ously from point to point, the strain at any point is 
defined by attending to the change which takes place 
in a very small sphere or cube having the point at its 
centre, so small that the strain throughout it may be 
regarded as homogeneous. In what follows we shall 
suppose strain to be homogeneous, unless the contrary is 
expressed. 



CHAP, v.] STRESS, STRAIN, AND RESILIENCE. 53 

49. The axes of a strain are the three directions in the 
body, at right angles to each other, which coincide with 
the directions of the axes of the ellipsoids. Lines drawn 
in the body in these three directions will remain at right 
angles to each other when the body is restored to its 
unstrained condition. 

A cube with its edges parallel to the axes will be 
altered by the strain into a rectangular parallelepiped. 
Any other cube will be changed into an oblique parallele- 
piped. 

When the axes have the same directions in space after 
as before the strain, the strain is said to be unaccompanied 
by rotation. When such parallelism does not exist, the 
strain is accompanied by rotation, namely, by the rotation 
which is necessary for bringing the axes from their initial 
to their final position. 

The numbers which specify a strain are mere ratios, 
and are therefore independent of units. 

50. When a body is under the action of forces which 
strain it, or tend to strain it; if we consider any plane 
section of the body, the portions of the body which it 
separates are pushing each other, pulling each other, or 
exerting some kind of force upon each other, across the 
section, and the mutual forces so exerted are equal and 
opposite. The specification of a stress must include a 
specification of these forces for all sections, and a body is 
said to be homogeneously stressed when these forces are 
the same in direction and intensity for all parallel sec- 
tions. We shall suppose stress to be homogeneous, in 
what follows, unless the contrary is expressed. 

51. When the force-action across a section consists of 



54 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

a simple pull or push normal to the section, the direction 
of this simple pull or push (in other words, the normal to 
the section) is called an axis of the stress. A stress (like 
a strain) has always three axes, which are at right angles 
to one another. The mutual forces across a section not 
perpendicular to one of the three axes are in general 
partly normal and partly tangential one side of the sec- 
tion is tending to slide past the other. 

The force per unit area which acts across any section is 
called the intensity of the stress on this section, or simply 
the stress on this section. The dimensions of "force per 

unit area," or - are , which we shall therefore call 

area LT 2 

the dimensions of stress. 

52. The relation between the stress acting upon a body 
and the strain produced depends upon the resilience of 
the body, which requires in general 21 numbers for its 
complete specification. When the body has exactly the 
same properties in all directions, 2 numbers are sufiicient. 
These specifying numbers are usually called coefficients of 
elasticity, but the word elasticity is used in so many 
senses that we prefer to call them coefficients of resilience. 
A coefficient of resilience expresses the quotient of a 
stress (of a given kind) by the strain (of a given kind) 
which it produces. A highly resilient body is a body 
which has large coefficients of resilience. Steel is an 
example of a body with large, and cork of a body with 
small, coefficients of resilience. 

In all cases (for solid bodies) equal and opposite strains 
(supposed small) require for their production equal and 
opposite stresses. 



v.] STRESS, STRAIN, AND RESILIENCE. 55 

53. The coefficients of resilience most frequently re- 
ferred to are the three following : 

(1) Resilience of volume, or resistence to hydrostatic 
compression. If V be the original and V - v the strained 

volume, is called the compression, and when the body 



is subjected to uniform normal pressure P per unit 
over its whole surface, the quotient of P by the compres- 
sion is the resilience of volume. This is the only kind of 
resilience possessed by liquids and gases. 

(2) Young's modulus, or the longitudinal resilience of 
a body which is perfectly free to expand or contract 
laterally. In general, longitudinal extension produces 
lateral contraction, and longitudinal compression produces 
lateral extension. Let the unstrained length be L and 

the strained length L I, then is taken as the measure 

JL 

of the longitudinal extension or compression. The stress 
on a cross section (that is, on a section to which the stress 
is normal) is called the longitudinal stress, and Young's 
modulus is the quotient of the longitudinal stress by the 
longitudinal extension or compression. If a wire of cross 
section A sq. cm. is stretched with a force of F dynes, 
and its length is thus altered from L to L + 1, the value 

Tjl T 

of Young's modulus for the wire is . . 

A. L 

(3) " Simple rigidity " or resistance to shearing. This 
requires a more detailed explanation. 

54. A shear may be denned as a strain by which a 
sphere of radius unity is converted into an ellipsoid of 
semiaxes 1, 1 + e, 1 - e ; in other words, it consists of an 



UNITS AND PHYSICAL CONSTANTS. [CHAP. 



extension in one direction combined with an equal com- 
pression in a perpendicular direction. 

55. A unit square (Fig. 1) whose diagonals coincide 
with these directions is altered by the strain into a 
rhombus whose diagonals are (1 + e) ^2 and (1 - e) ^/2, 
and whose area, being half the product of the diagonals, 



is 1 - 



to the first order of small quantities, is 1, 



the same as the area of the original square. The length 
of a side of the rhombus, being the square root of the 





Fig.1 



sum of the squares of the semi-diagonals, is found to be 
/\/l + e 2 or 1 + Je 2 , and is therefore, to the first order of 
small quantities, equal to a side of the original square. 

56. To find the magnitude of the small angle which a 
side of the rhombus makes with the corresponding side of 
the square, we may proceed as follows : Let acb (Fig. 2) 
be an enlarged representation of one of the small tri- 
angles in Fig. 1 . Then we have ab = J, cb = \e ^2 = ', 

A./ 2 



angle cba = -. Hence the length of the perpendicular cd 

is cb sin-= -- -- =^; and since ad is ultimately 

4 <s/2 \/2 2 

equal to ab, we have, to the first order of small quan- 
tities, 



v.] STRESS, STRAIN, AND RESILIENCE. 57 

angle cab= cd -^ e = e. 
ad \ 

The semi-angles of the rhombus are therefore e, 

and the angles of the rhombus are - '2e ; in other 

words, each angle of the square has been altered by the 
amount 2e. This quantity '2e is adopted as the measure 
of the shear. 

57. To find the perpendicular distance between oppo- 
site sides of the rhombus, we have to multiply a side by 
the cosine of 20, which, to the first order of small quan- 
tities, is 1. Hence the perpendicular distance between 
opposite sides of the square is not altered by the shear, 
and the relative movement of these sides is represented 



Fiff 3 - Fig .4 

by supposing one of them to remain fixed, while the 
other slides in the direction of its own length through a 
distance of 2e, as shown in Fig. 3 or Fig. 4. Fig. 3, in 
fact, represents a shear combined with right-handed rota- 
tion, and Fig. 4 a shear combined with left-handed rota- 
tion, as appears by comparing these figures with Fig. 1, 
which represents shear without rotation. 

58. The square and rhombus in these three figures may 
be regarded as sections of a prism whose edges are per- 
pendicular to the plane of the paper, and figures 3 and 4 



58 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

show that (neglecting rotation) a shear consists in the 
relative sliding of parallel planes without change of dis- 
tance, the amount of this sliding being proportional to the 
distance, and being in fact equal to the product of the 
distance by the numerical measure of the shear. A good 
illustration of a shear is obtained by taking a book, and 
making its leaves slide one upon another. 

It may be well to remark, by way of caution, that the 
selection of the planes is not arbitrary as far as direction 
is concerned. The only planes which are affected in the 
manner here described are the two sets of planes which 
make angles of 45 with the axes of the shear (these axes 
being identical with the diagonals in Fig. 1). 

59. Having thus defined and explained the term 
" shear," which it will be observed denotes a particular 
species of strain, we now proceed to define a shearing 
stress. 

A shearing stress may be defined as the combination of 
two longitudinal stresses at right angles to each other, 
these stresses being opposite in sign and equal in magni- 
tude ; in other words, it consists of a pull in one direction 
combined with an equal thrust in a 
D c perpendicular direction. 




60. Let P denote the intensity 
of each of these longitudinal 
stresses; we shall proceed to cal- 
culate the stress upon a plane in- 
-3 clined at 45 to the planes of these 
stresses. Consider a unit cube so 
taken that the pull is perpendicular 
to two of its faces, AB and DC (Fig. 5), and the thrust 



v.] STRESS, STRAIN, AND RESILIENCE. 5i> 

is perpendicular to two other faces, AD, BC. The forces 
which hold the half-cube ABC in equilibrium are 

(1) An outward force P, uniformly distributed over the 
face AB, and having for its resultant a single force P 
acting outward applied at the middle point of AB. 

(2) An inward force P, having for its resultant a single 
force P acting inwards at the middle point of BC. 

(3) A force applied to the face AC. 

To determine this third force, observe that the other 
two forces meet in a point, namely, the middle point of 
AC, that their components perpendicular to AC destroy 

one another, and that their components along AC, or 

p 

rather along CA, have each the magnitude - - . ; hence 

v^ 

their resultant is a force P ^2, tending from towards A. 

The force (3) must be equal and opposite to this. Hence 
each of the two half-cubes ABC, ADC exerts upon the 
other a force P ^2, which is tangential to their plane of 
separation. The stress upon the diagonal plane AC is 
therefore a purely tangential stress. To compute its 
intensity we must divide its amount P ^2 by the area of 
the plane, which is ^/2, and we obtain the quotient P. 
Similar reasoning applies to the other diagonal plane BD. 
P is taken as the measure of the shearing stress. The above 
discussion shows that it may be defined as the intensity of 
the stress either on the planes of purely normal stress, or 
on the planes of purely tangential stress. 

61. A shearing stress, if applied to a body which has 
the same properties in all directions (an isotropic body), 
produces a simple shear with the same axes as the stress ; 
for the extension in the direction of the pull will be equal 
to the compression in the direction of the thrust ; and in 



60 



UNITS AND PHYSICAL CONSTANTS. [CHAP. 



the third direction, which is perpendicular to both of 
these, there is neither extension nor contraction, since 
the transverse contraction due to the pull is equal to the 
transverse extension due to the thrust. 

A shearing stress applied to a body which has not the 
same properties in all directions produces in general a 
shear with the same axes as the stress, combined with 
some other distortion. 

In both cases, the quotient of the shearing stress by the 
shear produced is called the resistance to shearing. In the 
case of an isotropic body, it is also called the simple rigidity. 

62. The following values of the resilience of liquids 
under compression are reduced from those given in 
Jamin, "Cours de Physique," 2nd edition, torn. i. pp. 
168 and 169: 





Temp 
Cent. 


Coefficient of 
Resilience. 


Compression for 
one Atmosphere 
(megadyne per 
square centim.) 


Mercury, - 


o-o 


3'436xlO n 


2-91 x 10- 6 


Water, 


o-o 


2-02 xlO 10 


4-96 xlO-" 5 


. 


1-5 


1-97 


5-08 , 


. 


4-1 


2-03 


4-92 


. 


10-8 


2-11 


4-73 




13-4 


2-13 


4-70 , 


. 


13-0 


2-20 


4-55 , 


. 


25-0 


222 


4-50 




34-5 


224 


4-47 


. 


43-0 


2-29 


4-36 , 


. 


53-0 


2-30 


4-35 , 


Ether, - 


( ') 

o-o \ 

U4-OJ 


9'2 x 10 9 

7-8 

7'2 


l-09x!0- 4 
1-29 , 
1-38 


Alcohol, 


\ 7-3 | 
j 13-1 i 


1-22 xlO 10 
1-12 


8-17 
S-91 x 10-- r > 


Sea Water, - 


17'5 


2-33 


4-30 ,, 



STRESS, STRAIN, AND RESILIENCE. 



61 






63. The following are reduced from the results ob- 
tained by Amaury and Descamps, " Comptes Rendus," 
torn. Ixviii. p. 1564 (1869), and are probably more 
accurate than the foregoing, especially in the case of 
mercury : 







Coefficient of 
Resilience. 


Compression for 
one megadyne per 
square centim. 


Distilled Water, 


15 


2'22 x 10 10 


4-51 x 10- s 


Alcohol, 





1-21 


8-24 ,, 


. 


15 


I'll ,, 


8-99 ,, 


Ether, 





9'30xl0 9 


1-OSxlO- 1 




14 


7-92 


1-26 


Bisulphide of Carbon, 
Mercury, - 


14 
15 


l-60xl0 10 
5'42xlO n 


6-26xlO- 5 
l-84x!0- 6 



64. The following values of the coefficients of resilience 
for solids are reduced from those given in my own papers 
to the Royal Society (see " Phil. Trans.," Dec. 5th, 1867, 
p. 369), by employing the value of g at the place of ob- 
servation, namely, 981 '4. 





Young's 
Modulus, 


Simple 
Rigidity. 


Resilience of 
Volume, 


Density 


Glass, flint, 


6-03 xlO 11 


2-40 x 10 11 


4-15 xlO 11 


2-942 


Another specimen 


5-74 


2-35 


3-47 


2-935 


Brass, drawn, - 


1-075X10 1 - 


3-66 





8-471 


Steel, - 


2-139 ,, 


8-19 


1 -841x1 1 - 


7'849 


Iron, wrought, - 


1-963 ,, 


7-69 


1-456 


7-677 


cast,- 


1-349 


5-32 


9-64 xlO 11 


7-235 


Copper, - 


1-234 


4-47 


l-684x!0 12 


8-843 



65. The resilience of volume was not directly observed, 
but was calculated from the values of " Young's modulus " 
and " simple rigidity," by a formula which is strictly true 



62 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

for bodies which have the same properties in all direc- 
tions. The contraction of diameter in lateral directions 
for a body which is stretched by purely longitudinal stress 
was also calculated by a formula to which the same 
remark applies. The ratio of this lateral contraction to 
the longitudinal extension is called " Poisson's ratio," and 
the following were its values as thus calculated for the six 
bodies experimented on : 

Glass, flint, '258 Iron, wrought, '275 

Another specimen, "229 ,, cast, '267 

Brass, drawn, '469 (?) Copper, '378 

Steel, '310 

Kirchhoff has found for steel the value '294, and Clerk 
Maxwell has found for iron -267. Cornu ("Cornptes 
Rendus," August 2, 1869) has found for different speci- 
mens of glass the values -225, -226, '224, -257, '236, '243, 
250, giving a mean of '237, and maintains (with many 
other continental savants) that for all iso tropic solids 
(that is, solids having the same properties in all direc- 
tions) the true value is J. 

66. The following values of Poisson's ratio have been 
found by Mr. A. Mallock (" Proc. Roy. Soc.," June 19, 
1879) : 



Steel 


.... '253 


Ivory, 


.. -50 
.. '50 
50 


Brass 


325 


India Rubber, 
Paraffin, 


Copper, 


348 


Lead, . . ... 


.... '375 


Plaster of Paris,. . 
Cardboard, 
Cork, ... 


.. -181 
.. -2 
00 


Zinc (rolled),.... 

(CcLSt) 


.... '180 
230 


Ebonite, 


. '389 


Lial Longitudinal 
s to due to 
ndinal. Radial. 
2 -406 
3 '408 
86 '372 


In 

Cross 
Section, 

227 


Boxwood. 


Ra< 
due 

Longit 


Beechwood, 
White Pine, 


5 
-4 



v.] STRESS, STRAIN, AND RESILIENCE. 63 

The heading "Radial due to Longitudinal" means 
that the applied force is longitudinal (that is, parallel to 
the length of the tree) and that the contraction along a 
radius of the tree is compared with the longitudinal 
extension. 

67. The following are reduced from Sir W. Thomson's 
results ("Proc. Roy. Soc.," May, 1865), the value of g 
being 981 '4: 

Simple Rigidity. 
Brass, three specimens, 4'03 3*48 3'44 ) 1A11 

r* - J_ ' . A.Af\ A.4f\ f X 1U 



Copper, two specimens, 4*40 4*40 

Other specimens of copper in abnormal states gave 
results ranging from 3'86 x 10 11 to 4'64 x 10 11 . 

The following are reduced from Wertheirn's results 
("Ann. de Chim.," ser. 3, torn, xxiii.), g being taken as 
981: 

Different Specimens of Glass (Crystal}. 

Young's Modulus, 3 '41 to 4 '34, mean 3 '96 ^ 

Simple Rigidity, 1 -26 to 1 '66 1'48 [ x 10 11 

Volume Resilience 3'50 to 4*39 ,, 3'89j 

Different Specimens of Brass. 

Young's Modulus,.... 9*48 to 10 '44, mean 9 '86 \ 

Simple Rigidity, 3 '53 to 3 '90 ,, 3 '67 [ x 10 11 

Volume Resilience,.. 10'02tol0'85 ,, 10'43 J 

68. Savart's experiments on the torsion of brass wire 
(" Ann. de Chim.," 1829) lead to the value 3*61 x 10 11 for 
simple rigidity. 

Kupffer's values of Young's modulus for nine different 
specimens of brass range from 7 '9 6 x 10 11 to 11 '4 x 10 11 , 
the value generally increasing with the density. 

For a specimen, of density 8*4465, the value was 
10-58 x 10 11 . 



64 UNITS AND PHYSICAL CONSTANTS. [CHAP. v. 

For a specimen, of density 8 '4930, the value was 
11-2 x 10 11 . 

The values of Young's modulus found by the same experi- 
menter for steel, range from 20 -2 x 10 11 to 21 -4 x 10 11 . 

69. The following are reduced from Eankine's " Kules 
and Tables," pp. 195 and 196, the mean value being 
adopted where different values are given : 



Steel Bars, 


Tenacity. 
7'93x 10 9 


Y< >ung's Modulus. 
2-45 x 10 13 


Iron Wire, 


5-86 ,, 


1 '745 . 


Copper Wire, 
Brass Wire, 
Lead, Sheet, 
Tin, Cast, 


4-14 
3-38 

2-28 x 10 8 
3-17 


1-172 
9-81 x!0 1] 
5-0 xlO 10 


Zinc, 


5-17 ,, 




Ash,. 


ri72xlO !) 


1-10 x 10 11 


Spruce, 
Oak, 

Glass, 
Brick and Cement.. 


. 8-55 xlO 8 
1 -026x10'' 

6-48 x 10 s 
2-0 xl()< 


1-10 ,, 
1-02 

5-52 xlO 11 



The tenacity of a substance may be defined as the 
greatest longitudinal stress that it can bear without tear- 
ing asunder. The quotient of the tenacity by Young's 
modulus will therefore be the greatest longitudinal exten- 
sion that the substance can bear. 



65 



CHAPTER VI. 

ASTRONOMY. 

Size and Figure of the Earth. 

70. ACCORDING to the latest determination, as pub- 
lished by Capt. Clarke in the " Philosophical Magazine " 
for August, 1878, the semiaxes of the ellipsoid which 
most nearly agrees with the actual earth are, in feet, 

a = 20926629, b = 20925105, c = 20854477, 
which, reduced to centimetres, are 
a = 6-37839 x 10 8 , b = 6-37792 x 10 s , c = 6-35639 x 10 8 , 

giving a mean radius of 6 '3709 x 10 8 , and a volume of 
1-0832 x 10 27 cubic centims. 

The ellipticities of the two principal meridians are 

289-5 and 291T8' 

The longitude of the greatest axis is 8 15' W The mean 
length of a quadrant of the meridian is 1*00074 x 10 9 . 

The length of a minute of latitude is approximately 
185200 - 940 cos. 2 lat. of middle of arc. 

The mass of the earth, assuming Baily's value 5*67 for 
the mean density, is 6'14 x 10 27 grammes. 

E 



66 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

Day and Year. 

Sidereal day, ........................... 86164 mean solar seconds. 

Sidereal year, .......................... 31,558,150 

Tropical year, ......................... 31,556,929 

Angular velocity of earth's rotation, 



Velocity of earth in orbit, about 2960600 ,, 

Centrifugal force at equator due) O.OOAQ j 
to earth's rotation,.. ............. ) 3 3908 d y nes P er i ramme - 

Attraction, in Astronomy. 

71. The mass of the moon is the product of the earth's 
mass by -011364, and is therefore to be taken as 
6 '9 8 x 10 25 grammes, the doubtful element being the 
earth's mean density, which we take as 5 '67. 

The mean distance of the centres of gravity of the 
earth and moon is 60-2734 equatorial radii of the earth 
that is, 3-8439 x 10 10 centims. 

The mean distance of the sun from the earth is about 
1-487 x 10 13 centims., or 92*39 million miles, correspond- 
ing to a parallax of 8"-848.* 

The intensity of centrifugal force due to the earth's 

motion in its orbit (regarded as circular) is I ") r, r de- 

noting the mean distance, and T the length of the sidereal 
year, expressed in seconds. This is equal to the accelera- 
tion due to the sun's attraction at this distance. Putting 
for r and T their values, 1-487 x 10 13 and 3-1558 x 10 7 , 

we have / \ r= -5894. 

* This value of the mean solar parallax was determined by Pro- 
fessor Newcomb, and was adopted in the " Nautical Almanac " 
for 1882. (See Art. 86 for a later determination.) 



vi.] ASTRONOMY. 67 

This is about - - of the value of g at the earth's 
1660 

surface. 

The intensity of the earth's attraction at the mean dis- 
tance of the moon is about 

981 or -2701. 
(60-27) 2 

This is less than the intensity of the sun's attraction upon 
the earth and moon, which is "5894 as just found. Hence 
the moon's path is always concave towards the sun. 

72. The mutual attractive force F between two masses 
m and m', at distance I, is 

T? _ fl mm ' 

where C is a constant. To determine its value, consider 
the case of a gramme at the earth's surface, attracted by 
the earth. Then we have 

whence we find 

Cr= 6j48 = 1 

10 8 l-543x!0 7 ' 

To find the mass m which, at the distance of 1 centim. 
from an equal mass, would attract it with a force of 1 
dyne, we have 1 = C/n 2 ; 

whence m = I - = 3928 grammes. 

73. To find the acceleration a produced at the distance 
of I centims. by the attraction of a mass of m grammes, 

we have a = = C-, 

m' P 

where C has the value 6*48 x 10~ 8 as above. 



68 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

To find the dimensions of C we have C = , where the 

m 

dimensions of a are LT~ 2 . 

The dimensions of C are therefore 

L 2 M- 1 LT~ 2 ; that is, L 3 M' 1 T~ s . 

74. The equation = C^ shows that when a = l and 

1=1, m must equal ; that is to say, the mass which 
O 

produces unit acceleration at the distance of 1 centimetre 
is 1*543 x 10 7 grammes. If this were taken as the unit 
of mass, the centimetre and second being retained as the 
units of length and time, the acceleration produced by the 
attraction of any mass at any distance would be simply 
the quotient of the mass by the square of the distance. 

It is thus theoretically possible to base a general system 
of units upon two fundamental units alone ; one of the 
three fundamental units which we have hitherto employed 
being eliminated by means of the equation 

mass = acceleration x (distance) 2 , 

which gives for the dimensions of M the expression 
L 3 T- 2 . 

Such a system would be eminently convenient in astro- 
nomy, but could not be applied with accuracy to ordinary 
terrestrial purposes, because we can only roughly compare 
the earth's mass with the masses which we weigh in our 
balances. 

75. The mass of the earth on this system is the 
product of the acceleration due to gravity at the earth's 
surface, and the square of the earth's radius. This 

product is 

981 x (6-37 x!0 8 ) 2 = 3-98 x 10 20 , 



vi.] ASTRONOMY. 69 

and is independent of determinations of the earth's 
density. 

The new unit of force will be the force which, acting 
upon the new unit of mass, produces unit acceleration. 
It will therefore be equal to 1'543 x 10 7 dynes; and its 
dimensions will be 

mass x acceleration = (acceleration) 2 x (distance) 2 

= L 4 T -4 

76. If we adopt a new unit of length equal to I 
centims., and a new unit of time equal to t seconds, while 
we define the unit mass as that which produces unit 
acceleration at unit distance, the unit mass will be 
Pt~- x 1-543 x 10 7 grammes. 

If we make I the wave-length of the line F in vacuo, 
say, 4-86 x lO" 5 , 

and t the period of vibration of the same ray, so that 

is the velocity of light in vacuo, say, 
3 x 10 10 , 

the value of I s r' 2 or ill V fe 



4-374 x 10 16 , 

and the unit mass will be the product of this quantity 
into 1-543 x 10 7 grammes. This product is 6-75 x 10 23 
grammes. 

The mass of the earth in terms of this unit is 

3-98 x 10 20 4- (4-374 x 10 16 ) = 9100, 

and is independent of determinations of the earth's 
density. 



70 



CHAPTER VII. 
VELOCITY OF SOUND. 

77. THE propagation of sound through any medium is 
due to the elasticity or resilience of the medium ; and 
the general formula for the velocity of propagation s is 

E 



where D denotes the density of the medium, and E the 
coefficient of resilience. 

78. For air, or any gas, we are to understand by E the 
quotient 

increment of pressure 
corresponding compression ' 

that is to say, if P, P + p be the initial and final pres- 
sures, and V, V - v the initial and final volumes, p and v 
being small in comparison with P and V, we have 



V 

If the compression took place at constant temperature, 
we should have 



But in the propagation of sound, the compression is 
effected so rapidly that there is not time for any sensible 
part of the heat of compression to escape, and we have 



CHAP. VIL] VELOCITY OF SOUND. 71 



where y= 1-41 for dry air, oxygen, nitrogen, or hydrogen. 

p 
The value of =- for dry air at t Cent, (see p. 46) is 

(1 + -00366*) x 7-838 + 10 8 . 
Hence the velocity of sound through dry air is 
*= 10 4 ^1-41 x (1 + -00366*) x 7-838 

= 33240^1 + -00366*; 

or approximately, for atmospheric temperatures, 
= 33240 + 60*. 

79. In the case of any liquid, E denotes the resilience 
of volume.* 

For water at 8-l C. (the temperature of the Lake of 
Geneva in Colladon's experiment) we have 
E = 2-08 x 10 10 , D = 1 sensibly ; 

-= > /E = 144000, 
D 

the velocity as determined by Colladon was 143500. 

80. For the propagation of sound along a solid, in the 
form of a thin rod, wire, or pipe, which is free to expand 
or contract laterally, E must be taken as denoting Young's 
modulus of elasticity.* The values of E and D will be 
different for different specimens of the same material. 
Employing the values given in the Table ( 64), we have 

* Strictly speaking, E should be taken as denoting the resili- 
ence for sudden applications of stress so sudden that there is 
not time for changes of temperature produced by the stress to be 
sensibly diminished by conduction. This remark applies to both 
79 and 80. For the amount of these changes of temperature, 
see a later section under Heat. 



72 



UNITS AND PHYSICAL CONSTANTS. [CHAP. 













Values of E. 


Values of 


Values of / , 






D. 


or velocity. 


Glass, first specimen, 


6-03 xlO 11 


2-942 


4-53 x 10 5 


,, second specimen, 


5-74 


2-935 


4-42 


Brass, 


1-075x1 1 12 


8-471 


3-56 


Steel, 


2-139 


7-849 


5-22 


Iron, wrought, - 


1-963 


7-677 


5-06 


,, cast, - 


1-349 


7-235 


4-32 


Copper, 


1-234 ,, 


8-843 


3-74 



81. If the density of a specimen of red pine be -5, and 
its modulus of longitudinal elasticity be T6 x 10 6 pounds 
per square inch at a place where g is 981, compute the 
velocity of sound in the longitudinal direction. 

By the table of stress, page 4, a pound per square inch 
(g being 981) is 6*9 x 10 4 dynes per square centim. Hence 
we have for the required velocity 



centims. per second. 

82. The following numbers, multiplied by 10 5 , are the 
velocities of sound through the principal metals, as 
determined by Wertheim : 





At 20 C. 


At 100 C. 


At 200 C. 


Lead, 


1-23 


1-20 




Gold, 


1-74 


1-72 


1-73 


Silver, 


2-61 


2-64 


2-48 


Copper, - 
Platinum, 


3-56 
2-69 


3-29 

2-57 


2-95 
2-46 


Iron, 


5-13 


5-30 


4-72 


Iron Wire (ordinary), 
Cast Steel, 


4-92 
4-99 


5-10 
4-92 


4-79 


Steel Wire (English), 


4-71 


5-24 


5-00 





4-88 


5-01 





VII.] 



VELOCITY OF SOUND. 



73 



The following velocities in wood are from the observa- 
tions of Wertheim and Chevandier, " Comptes Rendus," 
1846, pp. 667 and 668 : 





., ., Radial Tangential 
Along Mbres. Direction . Direction. 


Pine, 






3-32 xO 5 ! 2-83 xlO 5 1 I'59xl0 5 


Beech, 




3-34 


3-67 i 2-83 


Witch-E 


m, - 


3-92 


3-41 ,, 2-39 


Birch, 






4-42 


2-14 ,, 3-03 ,, 


Fir, 






4-64 


2-67 1-57 


Acacia, 






4-71 




Aspen, 


- 




5-08 





Musical Strings. 

83. Let M denote the mass of a string per unit length, 
F stretching force, 

L ,, length of the vibrating portion ; 

then the velocity with which pulses travel along the 
string is 

F 



and the number of vibrations made per second is 



Example. 

For the four strings of a violin the values of M in 
grammes per centimetre of length are 

00416, -00669, -0106, -0266. 
The values of n are 



660, 440, 293J, 



195$; 



74 UNITS AND PHYSICAL CONSTANTS. [CHAP. VIK 

and the common value of L is 33 centims. Hence the 
values of v or 2Ln are 

43560, 29040, 19360, 12910 

centims. per second ; and the values of F or Mv 2 , in 
dynes, are 

7-89 x 10 6 , 5-64 x 10 8 , 3-97 x 10 6 , 4-43 x 10 6 . 

Faintest Audible Sound. 

84. Lord Rayleigh ("Proc. K. S.," 1877, vol. xxvi. p. 
248), from observing the greatest distance at which a, 
whistle giving about 2730 vibrations per second, and blown 
by water-power, was audible without effort in the middle 
of a fine still winter's day, calculates that the maximum 
velocity of the vibrating particles of air at this distance 
from the source was '0014 centims. per second, and that 
the amplitude was 8'1 x 10~ 8 centims., the calculation 
being made on the supposition that the sound spreads 
uniformly in hemispherical waves, and no deduction being 
made for dissipation, nor for waste energy in blowing. 



75 




CHAPTER VIII. 
LIGHT. 



85. ALL kinds of light are believed to have the same 
velocity in vacuo. The velocity of light of given re- 

frangibility in any medium is - of its velocity in vacuo, 

fji denoting the absolute index of refraction of that medium 
for light of the given refrangibility. 

Light of given refrangibility is light of given wave- 
frequency. Its wave-length in any medium is the 
quotient of its velocity in that medium by its wave- 
frequency. If n denote the wave-frequency (that is to 
say, the number of waves which traverse a given point 

in a second), the wave-length in any medium will be 

np 

of the velocity in vacuo. 

The absolute index of refraction for ordinary air is 
about 1*00029. More accurate statements of its value 
will be found in Arts. 94-96. 

86. The best determination of the velocity of light is 
that made by Professor Newcomb at Washington in 1882 
(" Astron. Papers of Amer. Ephem.," vol. ii. parts iii. 
and iv. 1885). The method employed was that of the 
revolving mirror, the distance between the revolving and 



76 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

the fixed mirror being in one portion of the observations 
2550 metres, and in the remaining portion 3720 metres. 
The resulting velocity in vacuo is 

2-99860 x 10 10 centims. per sec. 

The following summary of results is from Professor 
Newcomb's paper, page 202 : 

km. per. sec. 

Michelson, at Naval Academy, in 1879, 299910 

Michelson, at Cleveland, 1882, 299853 

Newcomb, at Washington, 1882, using only'j 

results supposed to be nearly free from 1- 299860 

constant errors, J 

Newcomb, including all determinations, 299810 

Foucault, at Paris, in 1862, 298000 

Cornu, at Paris, in 1874, 298500 

Cornu, at Paris, in 1878, 300400 

This last result as discussed by Listing, 299990 

Young and Forbes, 1880-81, 301382 

Professor Newcomb remarks (page 203) that the value 
299860 km. per sec. for the velocity of light, combined 
with Clark's value 6378-2 km. for the earth's equatorial 
radius, and Nyren's value 20"*492 for the constant of 
aberration, gives for the solar parallax the value 8" '7 94. 

87. The following are the wave-lengths adopted by 
Angstrom for the principal Fraunhofer lines in air at 760 
millims. pressure (at Upsal) and 16 C. : 

Centims. 

A 7-604 xlO~ 5 

B 6-867 

C 6-56201 ,, 

Mean of lines D 5-89212 ,, 

E 5-26913 

F 4-86072 

G 4-30725 

H 5 3-96801 

H 2 3-93300 



V 
viii.] LIGHT. 77 

These numbers will be approximately converted into 
the corresponding wave-lengths in vacuo by multiplying 
them by 1-00029. 

88. Assuming 3 x 10 10 to be the velocity of light in 
air, and neglecting the difference of velocity between 
the more and less refrangible rays, we obtain the follow- 
ing frequencies by dividing the common velocity by 
Angstrom's values of the wave-lengths : 

Vibrations per Second. 

A 3-945 xlO 14 

B 4-369 

C 4-572 

D 5-092 

E 5-693 ,, 

F 6-172 

G 6-965 

H x 7-560 ,, 

H 2 7-628 

According to Langley ("Com. Ren.," Jan., 1886), the 
solar spectrum extends beyond the red as far as wave- 
length 27 x 10~ 5 , and the radiation from terrestrial bodies 
at temperatures below 100 extends as far as wave-length 
150 x 10~ 5 . The frequencies corresponding to these two 
wave-lengths are 1*1 x 10 14 and 2 x 10 13 . 

INDICES OF REFRACTION OF SOLIDS. 

89. Dr. Hopkinson (" Proc. R. S.," June 14, 1877) has 
determined the indices of refraction of the principal 
varieties of optical glass made by Messrs. Chance, for the 
fixed lines A, B, C, D, E, 6, F, (G), G, h, H r By D is 
to be understood the more refrangible of the pair of 
sodium lines ; by b the most refrangible of the group of 
magnesium lines ; by (G) the hydrogen line near G. 



78 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

In connection with the results of observation, he 
employs the empirical formula 

/u,- 1 =a{l +bx(l + cx)}, 

where # is a numerical name for the definite ray of which 
ju is the refractive index. In assigning the value of x, 
four glasses hard crown, soft crown, light flint, and 
dense flint were selected on account of the good accord 
of their results ; and the mean of their indices for any 
given ray being denoted by /I, the value assigned to x for 
this ray is /i - /x p where /X F denotes the value of ji for the 
line F. 

The value of /x as a function of A., the wave-length in 
10~ 4 centimetres, was found to be approximately 

ji. = 1-538414 + 0-00676601 - 0-00017341 

+ 0-0(X)023l. 

The following were the results obtained for the different 
specimens of glass examined : 

Hard Crown, 1st specimen, density 2 '48575. 
= 0'523145, = 1-3077, c= -2'33. 

Means of observed values of /u. 

A 1-511755; B 1-513624; C 1 '514571 ; D 1-517116; 
E 1 -520324 ; b 1 '520962 ; F 1 '523145 ; (G) 1 '527996 ; 
O 1-528348; h 1-530904; H x 1-532789. 

Soft Crown, density 2' 55035. 
a=0 -5209904, 6 = 1-4034, c=-l'58. 

Means of observed values of /A. 

A 1-508956; B 1-510918; C 1-511910; D 1-514580; 
E 1-518017; 61'518678; F 1-520994; (G) 1 '526208; 
G 1-526592; h 1 '529360; Hj 1 '531415. 



vm.] LIGHT. 79 

Extra Light Flint Glass, density 2 '86636. 

a = 0'549123, 6 = 17064, c=-0'198. 

Means of observed values of /*. 

A 1-534067; B 1*536450; Cl '537682; 

D 1-541022; E 1 '545295; b 1 '546169; 

F 1-549125; (G) 1 '555870; G 1 '556375; 

7i 1-559992; Hj 1 '5627 60. 



Light Flint Glass, density 3 '20609. 

a= 0-583887, 6=1-9605, c=0'53. 

Means of observed values of u. 



B 1-568558; 
E 1-579227; 
<G) 1-592184; 
H, 1-600717. 


C 1-570007; 
b 1-580273; 
G 1-592825; 


D 1-574013 
F 1-583881 
h 1-597332 



Dense Flint, density 3 '65865. 

a = 0-634744, b = 2-2694, c = 1'48. 

Means of observed values of fi. 

B 1-615704; C 1-617477; D 1-622411 

E 1-628882; b 1-630208; F 1-634748 

<G) 1-645268; G 1 '64607 1 ; h 1 '651830 



Extra Dense Flint, density 3 '88947. 
a=0'664226, 6=2-4446, c=l87. 

Means of observed values of /x. 

A 1-639143; B 1*642894; C 1-644871 

D 1-650374; E 1-657631 ; b 1 '659 108 

F 1-664246; (G) 1-676090; G 1 '677020 

h 1-683575; H x 1-688590. 



80 UNITS AND PHYSICAL CONSTANTS. [CHAP, 

Double Extra Dense Flint, density 4 '42162. 
a=0'727237, fc=27690, c=2'70. 

Means of observed values of p. 

A 1-696531; B 1-701080; Cl "703485; 

D 1-710224; E 1 "719081 ; b 1 '720908; 

F 1-727257; (G) 1742058; G 1-743210; 

h 1-751485. 

90. The following indices of rock salt, sylvin, and alum 
for the chief Fraunhofer lines are from Stefan's observa 
tions : 

Rock Salt Sylvin Alum 

at 17 C. at 20 C. at 21 C. 

A 1-53663 1-48377 1-45057 

B -53918 -48597 '45262 

C -54050 -48713 -45359 

D -54418 -49031 -45601 

E -54901 -49455 -45892 

F -55324 -49830 -46140 

G -56129 -50542 -46563 

H -56823 -51061 -46907 

91. Indices of other singly refracting solids 



E 
Diamond, 


Index of 
.efraction. 

2-470 
1-4339 
1-532 
I 545 
1-528 
1-480 
1-593 
1-528 


Kind of 
Light. 

D 
D 
D 
Red 
Red 
Red 
D 
Red 


Observer. 

Schrauf. 
Stefan. 
Kohlrausch. 
Jamin. 
ii 
> j 
Baden Powell. 
Wollaston. 


Fluor-spar, 
Amber, 

Rosin 


Copal, . 


Gum Arabic, 
Peru Balsam, 
Canada Balsam,. 



Effect of Temperature. 

According to Stefan, the index of refraction of glass 
increases by about -000002 for each degree Cent, of 



VIII.] 



LIGHT. 



81 



increase of temperature, and the index of rock salt 
diminishes by about '000 037 for each degree of increase 
of temperature. 

92. Doubly refracting crystals : 
Uniaxal Crystals. 



Ice, 


Ordinary 
Index. 

1 -3060 


Extraordi- 
nary 
Index. 

1-3073 


Kind 
of 
Light. 

lied 


Tern] 


>. Observer. 
Reusch. 


Iceland-spar, 
Nitrate of Soda, 
Quartz, . 


1-65844 
1-5854 
1-54419 


1-48639 
1 -3369 
1-55329 


D 
D 
D 


24 

23 
24 


v. d. Willigen. 
F. Kohlrausch. 
v. d. Willigen. 


Tourmaline,... . 
Zircon, .. 


1-6479 
1-92 


1-6262 
1-97 


Green 
Red 


22 


Heusser. 
de Senarmont. 



Arragonite, 

Borax, 

Mica, 

Nitre, 

Selenite, 

Sulphur 



Biaxal Crystals. 
CIPAL INI 

Least. 
1-53013 
1-4463 
T5609 
1 -3346 
1-52082 



1-9505 



(prismatic)/ 
Topaz, 1-61161 



CES OF REFRACTION 


FOR SODIUM LIGHT. 


itermediate 
1-68157 
1-4682 


i. Greatest. 
1-68589 
1-4712 


Temp. 
23 


Observer. 
Rudberg. 
Kohlrausch. 


1-5941 


1-5997 


23 


,, 


1-5056 


1 -5064 


16 


Schrauf. 


1-52287 


1-53048 


17 


v. Lang. 


2-0383 


2-2405 


16 


Schrauf. 


1-61375 


1-62109 




Rudberg. 



INDICES OF REFRACTION FOR LIQUIDS. 

93. The following values of indices of refraction for 
liquids are condensed from Fraunhofer's determinations, 
as given by Sir John Herschel (" Enc. Met. Art," Light, 
p. 415):- 

Water, density 1 '000. 

B 1-3309; C T3317 ; D P3336; 

E 1-3358; F1'3378; G1'3413; 

H 1-3442. 



82 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

Oil of Turpentine, density 0'885. 

B 1 -4705 ; C 1 -4715 ; D 1 "4744 ; E 1 '4784 ; 
F 1-4817; G 1-4882; H 1-4939. 

The following determinations of the refractive indices 
of liquids are from Gladstone and Dale's results, as given 
in Watt's "Dictionary of Chemistry," iii. pp. 629-631 : 

Sidphide of Carbon, at temperature 11. 
A 1-6142; B 1-6207; C 1'6240; D 1-6333; 
E 1-6465; Fl'6584; G 1-6836; H 17090. 

Benzene, at temperature 10 "5. 

A 1-4879; B 1-4913; C 1-4931 ; D 1-4975; 
E 1-5036; F 1-5089; G 1-5202; H 1-5305. 

Chloroform, at temperature 10. 

A 1-4438; B 1'4457; C 1-4466; D 1-4490; 
E 1-4526; F 1-4555; G 1-4614; H 1-4661. 

Alcohol, at temperature 15. 

A 1-3600; B 1-3612; C 1'3621 ; D 1*3638; 
E 1-3661; F 1-3683; Gl'3720; H 1-3751. 

Ether, at temperature 15. 

A 1-3529; B 1'3545; C 1-3554; D 1-3566; 
E 1-3590; F 1-3606; G 1-3646; H 1-3683. 

Water, at temperature 15. 

A 1 -3284 ; B 1 -3300 ; C 1 '3307 ; D 1 -3324 ; 
E 1-3347; F 1-3366; G 1*3402; H 1-3431. 



INDICES FOR GASES. 

94. Indices of refraction of air at C. and 760 mm. 
for the principal fraunhofer lines. 



viii.] LIGHT. 83 

According to Kettler. According to Lorenz. 

A 1-00029286 1-00028935 

B 29350 28993 

C 29383 29024 

D 29470 29108 

K 29584 29217 

F 29685 29312 

G 29873 29486 

H 30026 29631 

95. The formula established by the experiments of 
Biot and Arago for the index of refraction of air at 
various pressures and temperatures was 

_ -0002943 h 

l+at "760' 

a denoting the coefficient of expansion '00366, and h the 
pressure in millims. of mercury at zero. As the pressure 
of 760 millims. of such mercury at Paris is 1-0136 x 10 
dynes per sq. cm., the general formula applicable to all 
localities alike will be 

_ l _ -0002943 P 

1 + -00366* ' l-0136x!0 6 ' 

where P denotes the pressure in dynes per sq. cm. This 
can be reduced to the form 

jt _ 1 _ -0002903 _P 
1 + -00366* ' 10* 

96. According to Mascart, /JL - 1 for any gas is pro- 
portional not to - - but to 

h + /3k' 2 

T+vT 

where /? and a are coefficients which vary from one gas 
to another. In the following table, the column headed /* 



84 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

contains the indices for and 760 mm. at Paris. The 
next column contains the value of /? multiplied by 10 7 (it 
being understood that h is expressed in millimetres), and 
the next column the value of a. All these data are for 
the light of a sodium flame : 

H> x 10 7 a' 

Air, 1-0002927 7 '2 '00382 

Nitrogen, 2977 8'5 382 

Oxygen, 2706 11-1 

Hydrogen, 1387 -8'6 378 

Nitrous Oxide, 5159 88 388 

NitrousGas, 2975 7 367 

Carbonic Oxide, 3350 8 '9 367 

Carbonic Acid, 4544 72 406 

Sulphurous Acid,.... 7036 25 460 

Cyanogen, 8216 27'7 

More recent, and probably more accurate observations, 
which will be published in vol. v. of " Travaux et 
Memoires du Bureau International des Poids et Mesures," 
1m ve been conducted by Benoit with Fizeau's dilatometer. 
They give 

1-0002923 

as the index of refraction of air for the I) line at C. 
wild 760 mm.; and for the temperature coefficient they 

give 

003667, 

whi.-h is identical with the coefficient of expansion of air. 
The larger value, -00382, obtained by Mascart, is traced 
to imperfect measurement of temperature. 

Coefficient of Dispersive Power. 
97. Assuming Cauchy's formula 




viii.] LIGHT. 85 

(where A is the wave-length), which is known to be 
approximately true for air within the limits of the visible 
spectrum, the constant b may be called the coefficient of 
dispersive power. Employing as the unit of length for 
A, the 10~* of a centimetre, Mascart ("Ann. de 1' Ecole 
Normale," 1877, p. 62) has obtained the following values 
for 6 : 

Coefficient of Dispersion. 

Air, -0058 

Nitrogen, '0067 

Oxygen, '0064 

Hydrogen, '0043 

Carbonic Oxide, '0075 

Carbonic Acid, '0052 

Nitrous Oxide, '0125 

Cyanogen, '0100 

According to Mascart, the ratio of dispersion to devia- 
tion for the two lines B and H is '024 for air, -032 for 
the ordinary ray in quartz, -038 for light crown glass, 
040 for water, and '046 for the ordinary ray in Iceland- 
spar. 

Rotation of Plane of Polarization. 

98. The rotation produced by 1 millim. of thickness of 
quartz cut perpendicular to the axis has the following 
values for different portions of the spectrum, according to 
the observations of Soret and Sarasin (" Com. Ren. 95," 
p. 635, 1882), the temperature of the quartz being 
20 C. : 



Rotation. 

A 12-668 

B 15-746 

C 17'318 

D 2 21-684 

D 21-727 



Rotation. 

E 27'543 

F 32773 

G 42 604 

H... , 51-193 



86 UNITS AND PHYSICAL CONSTANTS. [CHAP. vin. 

According to the same observers, the rotation at 
t C. is .equal to the rotation at C. multiplied by 
1+-000179*. 

Units of Illuminating Power. 

99. The British "Candle" is a spermaceti candle, 
J inch in diameter (6 to the lb.), burning 120 grains 
per hour. 

The French " Carcel " is a lamp of specified construc- 
tion, burning 42 grammes of pure Colza oil per hour. 
One "carcel" is equal to about 9J "candles." 

The unit adopted by the International Congress at 
Paris, April 1884, is a square centimetre of molten 
platinum at the temperature of solidification. The surface 
illuminated by it in photometric tests is to be normally 
opposite to the surface of the molten platinum. Accord- 
ing to the experiments of M. Violle the author of this 
unit, it is equal to 2 '08 carcels. It is therefore about 
19| candles. 



87 



CHAPTER IX. 
HEAT. 

100. THE unit of lieat is usually defined as the quantity 
of heat required to raise, by one degree, the temperature 
of unit mass of water, initially at a certain standard tem- 
perature. The standard temperature usually employed is 
C. ; but this is liable to the objection that ice may be 
present in water at this temperature. Hence 4 C. has 
been proposed as the standard temperature ; and another 
proposition is to employ as the unit of heat one hundredth 
part of the heat required to raise the unit mass of water 
from to 100 C. 

101. According to Regnault (" Mem. Acad. Sciences," 
xxi. p. 729) the quantity of heat required to raise a given 
mass of water from to t 0. is proportional to 

t + -000 02* 2 + -000 000 3* 3 . . . . (1) 

The mean thermal capacity of a body between two stated 
temperatures is the quantity of heat required to raise it 
from the lower of these temperatures to the higher, 
divided by the difference of the temperatures. The mean 
thermal capacity of a given mass of water between 0. 
and t is therefore proportional to 

1 + -000 02* + -000 000 3* 2 . ... (2) 



88 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

The thermal capacity of a body at a stated temperature 
is the limiting value of the mean thermal capacity as the 
range is indefinitely diminished. Hence the thermal 
capacity of a given mass of water at t is proportional to 
the differential coefficient of (1), that is to 

1 + -000 04* + -000 000 9* 2 . ... (3) 

Hence the thermal capacities at and 4 are as 1 to 
1-000174 nearly; and the thermal capacity at is to 
the mean thermal capacity between and 100 as 1 to 
1-005. 

102. If we agree to adopt the capacity of unit mass of 
water at a stated temperature as the unit of capacity, the 
unit of heat must be defined as n times the quantity of 
heat required to raise unit mass of water from this initial 

temperature through - of a degree when n is indefinitely 
n 

great. 

Supposing the standard temperature and the length of 
the degree of temperature to be fixed, the units both of 
heat and of thermal capacity vary directly as the unit of 
mass. 

In what follows, we adopt as the unit of heat (except 
where the contrary is stated) the heat required to raise 
a gramme of pure water through 1 C. at a temperature 
intermediate between and 4. This specification is 
sufficiently precise for the statement of any thermal 
measurements hitherto made. 

103. The thermal capacity of unit mass of a substance 
at any temperature is called the specific heat of the sub- 
stance at that temperature 



ix.] HEAT. 89 

Specific heat is of zero dimensions in length, mass, and 
time. It is in fact the ratio 

increment of heat in the substance 

increment of heat in water 

for a given increment of temperature, the comparison 
being between equal masses of the substance at the actual 
temperature and of water at the standard temperature. 
The numerical value of a given concrete specific heat 
merely depends upon the standard temperature at which 
the specific heat of water is called unity. 

104. The thermal capacity of unit volume of a sub- 
stance is another important element : we shall denote it 
by c. Let s denote the specific heat, and d the density of 
the substance ; then c is the thermal capacity of d units 
of mass, and therefore c = sd. The dimensions of c in 
length, mass, and time are the same as those of d, namely, 

. Its numerical value will not be altered by any change 

in the units of length, mass, and time, which leaves the 
value of the density of water unchanged. 

In the O.G.S. system, since the density of water 
between and 4 is very approximately unity, the 
thermal capacity of unit volume of a substance is the 
value of the ratio 

increment of heat in the substance 

increment of heat in water 

for a given increment of temperature, when the compari- 
son is between equal volumes. 

105. Mr. Herbert Tomlinson (" Proc. Roy. Soc.," June 
19, 1885) has obtained the following determinations of 
specific heat from observations conducted in a uniform 



90 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

manner with metallic wires well annealed. The wires 
were heated sometimes to 60 C. and sometimes to 100 
C., and were plunged in water at 20. The formula? are 
for the true specific heat at t C : 

Aluminium, '20700 + '0002304* 

Copper, -09008+ '0000648* 

German Silver, "09413 + '0000106* 

Iron, -10601 + '000 140* 

Lead, : '02998+ '000031* 

Platinum, '03198 + '000013* 

Platinum Silver, -04726 + '000028* 

Silver, -05466 + '000044? 

Tin, -05231 + '000072* 

Zinc, '09009+ '000075* 

The formula? for the mean specific heat between and t* 
are obtained from these by leaving the first term un- 
changed and halving the second term. 

Yiolle has made the following determinations of specific 
heat at t : 

Platinum, '0317+ '000012* 

Iridium, '0317+ '000012* 

Palladium, -0582 + '000020* 

H. F. Weber has determined the specific heat of 
diamond to be 

0947 + -000 994* - -000 000 3G* 2 , 

and consequently the mean specific heat of diamond from 
to t to be 

0947 + -000 497* - -000 000 12* 2 . 
The mean specific heat of ice according to Regnault is 
504 between - 20 and 0, and '474 between - 78 and 0. 
106. The following list of specific heats of elementary 
substances is condensed from that given in Landolt and 
Bornstein's tables : 



II.] 



HEAT. 



91 






Substance. 


Temperature. 


Sp. Heat. 


Observer. 


Aluminium, 


15 to 97 


2122 


Regnault. 


Antimony, 


. 13 106 


0486 


Bede. 


Arsenic (crystalline), 


. 21 68 


0830 


( Bettendorff 
j Wullner. 


,, (amorphous), 


. 21 65 


0758 


" 


Bismuth, , 


9 102 


0298 


Bede. 


Borax (crystalline), , 


100 


2518 


Mixter&Dana 


, , (amorphous), 


13 48 


254 


Kopp. 


Bromine, solid, 


-78 ,,-20 


0843 


Regnault. 


,, liquid, 


13 45 


1071 


Andrews. 


Cadmium, 


100 


0548 


Bunsen. 


Calcium, 


100 


1804 





Carbon, diamond, 


11 


112S 


H. F. Weber. 


graphite, 


11 


1604 


,, 


,, wood charcoal,.. 


to 99 


1935 


> 


Cobalt, 


9 97 


1067 


Regnault. 


Copper, 


15 100 


0933 


Bede. 


Gold, 


100 


0316 


Violle. 


Iodine, 


9 ,, 98 


0541 


Regnault. 


Iridium, 


100 


0323 


Violle. 


Iron, 


50 


1124 


Bystrom. 


Lead, 


19 to 48 


0315 


Kopp. 


Lithium, 


27 99 


9408 


Regnault. 


Magnesium , 


20 51 


245 


Kopp. 


Manganese, 


14 ,, 97 


1217 


Regnault. 


Mercury, solid, 


-78 ,,-40 


0319 




,, liquid, 


17 48 


0335 


Kopp. 


Molybdenum, 


5 15 


0659 


( De la Rive and 
( Marcet. 


Nickel, 


14 97 


1092 


Regnault. 


Osmium, 


19 98 


0311 


5 


Palladium, 


100 


0592 


Violle. 


Phosphorus ( yellow , solid ) 


-78 10 


1699 


Regnault. 


( ,, liquid) 


49 98 


2045 


Person. 


(red), 


15 98 


1698 


Regnault. 


Platinum, 


,, 100 


0323 


Violle. 


Potassium, 


-78 


1655 


Regnault. 



UNITS AND PHYSICAL CONSTANTS. [CHAP. 



Substance. 


Temperature. 


Sp. Heat. 


Observer. 


Rhodium, 


10 97 


0580 


Regnault. 


Selenium, crystalline, 


22 62 


0840 


j Bettendorff & 
j Wlillner. 


Silicon, crystalline, 


22 


1697 


H. F. Weber. 


Silver, 


to 100 


0559 


Bunsen. 


Sodium, 


-28 6 


2934 


Regnault. 


Sulphur (rhomb, cryst.), 


17 45 


163 


Kopp. 


,, (newly melted), 


15 97 


1844 


Regnault. 


Tellurium, crystalline, 


21 51 


0475 


Kopp. 


Thallium, 


17 100 


0335 


Regnault. 


Tin, cast, 


100 


0559 


Bunsen. 


Zinc, 


0",, 100 


0935 


J5 



Substances not Elementary. 
Brass (4 copper 1 tin), hard, 15 to 98 '0858 Regnault. 

soft, 14 98 '0862 
Ice, -20,, '504 

107. The following determinations of specific heat of 
liquids are by Regnault. We have omitted decimal 
figures after the fourth, as even the second figure is 
different with different observers : 



Alcohol. 
Temp. Sp. Ht. 
-20 -5053 


Chloroform. 
Temp. 8p. Ht. 

- 30 -2293 


Oil of Turpentine. 
Temp. Sp. Ht. 
-20 -3842 





5475 





2324 





4106 


40 


6479 


30 


2354 


40 


4538 


80 


7694 


60 


2384 


80 


4842 










120 


5019 










160 


5068 


Ether. 
Temp. Sp. Ht. 
-30 -5113 


Bisulphide of Carbon. 
Temp. Sp. Ht. 
-30 -2303 







5290 





2352 






30 


5468 


30 


2401 





Schiiller has found the specific heat of liquid benzine at 

to be 

37980 + -00144*. 



IX.] 



HEAT. 



93 



108. The following table (from Miller's "Chemical 
Physics," p. 308) contains the results of Regnault's ex- 
periments on the specific heat of gases. The column 
headed " equal weights " contains the specific heats in the 
sense in which we have defined that term. The column 
headed "equal volumes" gives the relative thermal capa- 
cities of equal volumes at the same pressure and tem- 
perature : 

Thermal Capacities of Gases and Vapours at 
Constant Pressure. 



Gas or Vapour. 


Equal 


Gas or Vapour. 


Equal 


Vols. 


Weights. 


Vols. 


Weights 


Air, - - - 
Oxygen, - 


2375 
2405 


2375 
2175 


Hydrochloric \ 
Acid, - -/ 


2352 


1842 


Nitrogen, - 
Hydrogen, 


2368 
2359 


2438 
3-4090 


Sulphuretted \ 
Hydrogen, / 


2S57 


2432 


Chlorine, - 


2964 


1210 


Steam, 


2969 


4805 


Bromine, - 


3040 


0555 


Alcohol, - 


7171 


4584 


Nitrous Oxide, 


3447 


2262 


Wood Spirit, - 


5063 


4580 


Nitric Oxide, - 


2406 


2317 


Ether, - - j 1 -2-266 


4796 


Carbonic Oxide 


2370 


2450 


Ethyl Chloride, 


6096 


2738 


Carbonic ) 
Anhydride, / 
Carbonic Di- \ 
sulphide, / 


3307 
4122 


2163 
1569 


,, Bromide, 
Disul- \ 
phide,/ 
,, Cyanide, 


7026 
1-2466 
8293 


1896 
4008 
4261 


Ammonia, 


2996 


5084 


Chloroform, - 


0401 


1566 


Marsh Gas, 


3277 


5929 


Dutch Liquid, 


7911 


2293 


OlefiantGas, - 


4106 


4040 


Acetic Ether, - 


1 -2184 


4008 


Arsenious \ 
Chloride, J 


7013 


1122 


Benzol, 
Acetone, - 


1-0114 
8341 


:^7o4 
4125 


Silicic Chloride 
Titanic 


7778 
8564 


1322 
1290 


OilofTurpen-| 
tine, - - J 


2-3776 


5061 


Stannic ,, 
Sulphurous \ 
Anhydride, J 


8639 
341 


0939 
154 


Phosphorus \ 
Chloride, / 


63b6 


1347 



94 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

109. E. Wiedemann (" Pogg. Ann.," 1876, No. 1, p. 
39) has made the following determinations of the specific 
heats of gases : 

Specific Heat. 

AtO. At 100. At 200. Jjjjg 

Air, 0-2389 ... ... 1 

Hydrogen, 3 '410 ... ... 0*0692 

Carbonic Oxide, 0-2426 ... ... 0*967 

Carbonic Acid, 0*1952 0*2169 0'23S7 1-529 

Ethyl, 0-3364 0-4189 0'5015 0'9677 

Nitric Oxide, 0*1983 0*2212 0*2442 1*5241 

Ammonia, 0*5009 0*5317 0*5629 0*5894 

Multiplying the specific heat by the relative density, 
he obtains the following values of 

Thermal Capacity of Equal Volumes. 

At O c . At 100. At 200. 

Air, 0*2389 

Hydrogen, 0*2359 

Carbonic Oxide,.. 0*2346 

Carbonic Acid,... 0*2985 0*3316 0*3650 

Ethyl, 0*3254 0*4052 0*4851 

Nitric Oxide, 0*3014 0*3362 0*3712 

Ammonia, 0*2952 0*3134 0*3318 

The same author ("Pogg. Ann.," 1877, New Series, 
vol. ii. p. 195) has made the following determinations of 
specific heats of vapours at temperature t : 

^P-. $S$SBL Specific Heat. 

Chloroform, -J6-9 to 189*8 -1341 + '0001354* 

Bromic Ethyl, . . 27 *9 to 1 89 ' * 1 354 + *003560 

Benzine, 34 '1 to 115 '1 '2237 + '0010228* 

Acetone, 26*2 to 179*3 "2984 + *0007738/ 

Acetic Ether, . . . 32 *9 to 1 1 3 *4 '2738 + *OOOS700 

Ether, 25 *4 to 188 *8 "3725 + '0008536/ 



ix.j HEAT. 95 

Regnault's determinations for the same vapours were 
as follows : 

Mean Specific Heat for this Range. 

Vanour Range of , *- . ^ 

Temperature. According to According to 

Regnault. Wiedemann. 

Chloroform, 117 to 228 '1567 '1573 

Bromic Ethyl,... 77 7 to 196 '5 "1896 -1841 

Benzine, 116 to 218 '3754 "3946 

Acetone, 129 to 233 "4125 '3946 

Acetic Ether,.... 115 to 219 '4008 '4190 

Ether, 70 to 220 -4797 "4943 

Regnault has also determined the mean specific heat of 
bisulphide of carbon vapour between 80 and 147 to be 
1534, and between 80 and 229 to be -1613. 

MELTING POINTS AND HEAT OF LIQUEFACTION. 
110. Violle has made the following determinations of 
melting points (" Com. Ren.," Ixxxix. p. 702) : 



Silver, 954 

Gold, 1045 

Copper, 1054 



Palladium, 1500 

Platinum, 1775 

Iridium, 1950 



This last temperature 1950 is very near to that of the 
hottest part of the oxyhydrogen flame. 

The same observer has found the latent heat of lique- 
faction of platinum to be 27 '2, and of palladium 36-3 
("Com. Ren." Ixxxv. p. 543, and Ixxxvii. p. 981). 

111. The following approximate table of melting points 
is based on that given in the second supplement to Watt's 
" Dictionary of Chemistry," pp. 242, 243 : 

Platinum, 2000 Copper, 1090 

Palladium, 1950 Silver, 1000 

Gold, 1200 Borax, 1000 

Cast Iron, 1200 Antimony, 432 

Glass, 1100 Zinc, 360 



96 



UNITS AND PHYSICAL CONSTANTS. [CHAP. 



Lead, 330 

Cadmium, 320 

Bismuth, 265 

Tin, 230 

Selenium, 217 

Cane Sugar, 160 

Sulphur, Ill 

Sodium, .. 



Wax, 68 

Potassium, 58 

Paraffin, 54 

Spermaceti, 44 

Phosphorus, 43 

Water, 

Bromine, -21 

Mercury, -40 



Mercury, 
Phosphorus, . . 
Lead, 


Melting 
Point. 
-39 
44 
332 


Latent 
Heat. 
2-82 
5-0 
5-4 


Sulphur, 
Iodine, 


115 
107 


9-4 
117 


Bismuth, 
Cadmium, 
Tin, .. 


270 
320 
235 


12-6 
13-6 
14-25 



Sodium, 90 

112. The following table (from Watt's "Dictionary of 
Chemistry," vol. iii. p. 77) exhibits the latent heats of 
fluidity of certain substances, together with their melting 
points : 

Melting Latent 
Point. Heat. 

Silver, 1000 21 '1 

Zinc, 433 28'1 

Chloride of Calcium 

(CaC1.3H 2 0),.... 28-5 40'7 
Nitrate of Potas- 
sium, 339 47'4 

Nitrate of Sodium, 310'5 63 '0 

The latent heat of fluidity of water was found by 
Regnault, and by Provostaye and Desains, to be 79. 
Bunsen, by means of his ice-calorimeter (" Pogg. Ann.,' 7 
vol. cxli. p. 30), has obtained the value 80-025. He 
finds the specific gravity of ice to be '9167. 

113. Chandler Roberts and Wrightson have compared 
the densities of molten and solid metals by weighing a 
solid metal ball in a bath of molten metal either of the 
same or a different kind (" Phys. Soc.," 1881, p. 195, and 
1882, p. 102). They find that "iron expands rapidly (as 
much as 6 per cent.) in cooling from the liquid to the 
plastic state, and then contracts 7 per cent, to solidity ; 
whereas bismuth appears to expand in cooling from the 
liquid to the solid state about 2 -35 per cent." The 
following is their tabular statement of results : 



IX.] 



HEAT. 



97 



Metal. 
Bismuth 


Sp. Grav. 
of Solid. 

9 -82 


Sp. Grav. 
of Liquid. 

10-055 


Percentage of change in 
volume from cold solid 
to liquid. 

Decrease of vol. 2*3 


Copper, ... 


8-8 


8-217 


Increase of vol. 7'1 


Lead 


11 '4 


10-37 


,, 9-93 


Tin, 


. . . 7'5 


7-025 


6-76 


Zinc, 


7-2 


6-48 


11-1 


Silver. .. 


,. 10-57 


9-51 


11-2 



Irpn (No. 4 foundry, ) fi . q5 fi . 8g 

Cleveland), j b 



1-02 



114. Change of volume in melting, from Kopp's experi- 
ments (Watt's "Die.," Art. Heat, p. 78) : 

Phosphorus. Calling the volume at unity, the volume at the 

melting point (44) is 1*017 in the solid, and 1-052 in the 

liquid, state. 
Sulphur. Volume at being 1, volume at the melting point 

(115) is 1-096 in the solid, and 1'150 in the liquid, state. 
Wax. Volume at being 1, volume at melting point (64) is 

1-161 in solid, and 1-166 in liquid, state. 
Stearic Acid. Volume at being 1, volume at melting point 

(70) is 1-079 in solid, and 1-198 in liquid, state. 
Ro*e? 8 Fusible Metal (2 parts bismuth, 1 tin, 1 lead). Volume at 

being 1, volume at 59 is a maximum, and is 1*0027. 

Volume at melting point (between 95 and 98) is greater 

in liquid than in solid state by 1 '55 per cent. 

115. The following table (from Miller's "Chemical 
Physics," p. 344) exhibits the change of volume of certain 
substances in passing from the liquid to the vaporous 
condition under the pressure of one atmosphere : 

1 volume of water yields 1696 volumes of vapour. 
,, alcohol 528 

ether 298 

,, oil of turpentine 193 ,, ,. 

G 



98 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

116. The following table of boiling points and heats of 
vaporization, at atmospheric pressure, is condensed from 
Landolt and Bornstein, pp. 189, 190 : 

Boiling Latent Heat of observer 
Point. Vaporization. Observer. 

Alcohol, 77'9 202-4 Andrews. 

Bisulphide of Carbon, 46 '2 86 '7 

Bromine, 58 45'6 ,, 

Ether, 34'9 90'4 

Mercury, 350 62'0 Person. 

Oil of Turpentine, 159'3 74 '0 Brix. 

Sulphur,.. 316 362'0 Person. 

Water, 100 535'9 Andrews. 

117. Regnault's approximate formula for what he calls 
"the total heat of steam at t" that is, for the heat 
required to raise unit mass of water from to t in the 
liquid state and then convert it into steam at t, is 

606-5 + -3052. 

If the specific heat of water were the same at all tempera- 
tures, this would give 

606-5 --695* 

as the heat of evaporation at t. But since, according 
to Regnault, the heat required to raise the water from 
to t is 

*+-00002 2 + -000 000 3* 3 , 

the heat of evaporation will be the difference between 
this and the " total heat," that is, will be 

606-5 - -695* - -000 02* 2 - -000 000 3* 3 , 
which is accordingly the value adopted by Regnault as 
the heat of evaporation of water at t. 

118. According to Regnault, the increase of pressure 
at constant volume, and increase of volume at constant 



ix.] HEAT. 99 

pressure, when the temperature increases from to 100, 
have the following values for the gases named : 



Gas. 



At Constant At Constant 

Volume. Pressure. 



Hydrogen "3667 '3661 

Air, -3665 '3670 

Nitrogen, '3668 

Carbonic Oxide, '3667 '3669 

Carbonic Acid, '3688 '3710 

Nitrous Oxide, '3676 '3719 

Sulphurous Acid, -3845 '3903 

Cyanogen, '3829 '3877 

Jolly has obtained the following values for the coeffi- 
cient of increase of pressure at constant volume : 

Air, -00366957 

Oxygen, -00367430 

Hydrogen, '00365620 

Nitrogen, '0036677 

Carbonic Acid, '0037060 

Nitrous Oxide, '0037067 

Mendelejetf and Kaiander have determined the co- 
efficient of expansion of air at constant pressure to be 
0036843. 

119. Regnault's results as to the departures from Boyle's 
law are given in the form 

Ii = lA(m-l)B(m-l)2, 
v o r o 
Vj denoting the volume at the pressure P p V the volume 

TjT 

at atmospheric pressure P . and m the ratio . 

^i 

For air, the negative sign is prefixed to A and the posi- 
tive sign to B, and we have 

log A = 3-0435120, 
log 6 = 5-2873751. 



100 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

For nitrogen, the signs are the same as for air, and we 
have 

log A = 4-8399375, 

log B = 6-8476020. 

For carbonic acid, the negative sign is to be prefixed 
both to A and B, and we have 

log A = 3-9310399, 

log B- 6-8624721. 

For hydrogen, the positive sign is to be prefixed both to 
A and B, and we have 

log A = 4-7381736, 

log B = 6-9250787. 

120. The following table, showing the maximum pres- 
sure of aqueous vapour at temperatures near the ordinary 
boiling point, is based on Regnault's determinations, as 
revised by Moritz (Guyot's Tables, second edition, collec- 
tion D, table xxv. ) : 

Centims. of TV*.* >r 

Temperature. Mercury * 

99-0 73-319 9-779 x 10 5 

99-1 73-584 9'814 

99-2 73-849 9 -849 ,, 

99-3 74-115 9-885 ,, 

99-4 74-382 9 '920 ,, 

99-5 74-650 9 '956 

99-6 74-918 9 "992 

997 75-187 1 -0028 x 10 s 

99-8 75-457 1-0064 

99-9 75-728 1-0100 ,, 

100-0 76-000 1-0136 

100-1 76-273 1-0173 ,, 

100-2 76-546 1-0209 ,, 

100-3 76-820 1-0245 ,, 



IX.] 



HEAT. 



101 



Temperature. 

100-4 
100-5 
100-6 
100-7 
100-8 
100-9 
101-0 



Centims. of 
Mercury 
at Paris. 

77-095 
77-371 
77-647 
77-925 
78-203 
78-482 
78-762 



Dynes per 
sq. cm. 

l-0282x!0 6 

1-0319 

1-0356 ,, 

1-0393 ,, 

1-0430 

1-0467 

1-0505 , 



121. Maximum Pressure of Aqueous Vapour at various 
temperatures, in dynes per sq. centim. 



-20 

15 

-10 

- 5 



5 

10 

15 



1236 

1866 

2790 

4150 

6133 

8710 

12220 

16930 



20 23190 

25 31400 

30 42050 

40 73200 



50 l-226x!0 5 

60 1-985 

80 4-729 

100 l-014xl0 6 

120 1-988 

140 3-626 

160 6-210 

180 l-006xl0 7 

200 1 -560 



Maximum Pressure of various Vapour s, in dynes per sq. cm. 





Alcohol. 


Ether. 


Sulphide of 
Carbon. 


Chloroform. 


-20 


4455 


9-19 x 10 4 


6-31 x 10 4 




-10 


8630 


1-53 x 10 5 


1-058 x 10 5 







16940 


2-46 


1-706 




10 


32310 


3-826 


2-648 




20 


59310 


5-772 


3-975 


2-141 x 10 5 


30 


1-048 x 10 5 


8-468 


5-799 


3-301 


40 


1-783 , 


1-210 x 10 6 


8-240 


4927 


50 


2-932 , 


1-687 


1-144 x 10 6 


7-14 


60 


4-671 , 


2-301 


1-554 


1-007 x 10 6 


80 


1-084 x 10 6 


4-031 


2-711 


1-878 


100 


2-265 , 


6-608 


4-435 


3-24 


120 


4-31 


1-029 x 10 7 


6-87 


5-24 



102 



UNITS AND PHYSICAL CONSTANTS. [CHAP. 



122. The following are approximate values of the 
maximum pressure of aqueous vapour at various tempera- 
tures, in millimetres of mercury. They can be reduced 
to dynes per sq. cm. by multiplying by 133*4 : 





mm. 




mm. 




mm. 




mm. 





4-6 


92 


567 


112 


1150 


132 


2155 


10 


9-2 


94 


611 


114 


1228 


134 


2286 


20 


17-4 


96 


658 


116 


1311 


136 


2423 


30 


31-fi 


98 


707 


118 


1399 


138 


2567 


40 


54-9 


100 


760 


120 


1491 


140 


2718 


50 


96-2 


102 


816 


122 


1588 


142 


2875 


60 


149 


- 104 


875 


124 


1691 


144 


3040 


70 


233 


106 


938 


126 


1798 


146 


3213 


80 


355 


108 


1004 


128 


1911 


148 


3393 


90 


525 


110 


1075 


130 


2030 


150 


3581 



123. The density (in gm. per cub. cm.) of aqueous vapour 
at any temperature t and any pressure p (dynesper sq. cm.), 
whether equal to or less than the maximum, pressure, is 

622 x -001276 y p 

1+-00366* X 106' , 

If q denote the pressure in millimetres of mercury, the 
approximate formula is 

622 x -001 293 q 
1 x -00366(5 X 760* 

124. Temperature of evaporation and dew-point 
(Glaisher's Tables, second edition, page iv.). The fol- 
lowing are the factors by which it is necessary to mul- 
tiply the excess of the reading of the dry thermometer 
over that of the wet, to give the excess of the tempera- 
ture of the air above that of the dew-point : 



Beading of 

Dry Bulb 

Therm. 

-10C.=14F. 
- 5 23 
32 
+ 5 41 
+ 10 50 



Factor. 

8-76 
7'28 
3-32 
2-26 
2-06 



Beading of 
Dry Bulb 
Therm. 

15C.=:59 F. 

20 68 

25 77 

30 86 

35 95 



Factor. 

1-89 
1-79 
T70 
1-65 
1-60 



IX.] 



HEAT. 



103 



125. Critical temperatures of gases, above which they 
cannot be liquefied (abridged from Landolt and Bornstein, 
p. 62) :- 



Critical 
mperature. 


Max. Pressure 
of Gas at this 
Temp. 


-174-2 


98'9atm. 


- 105-4 


48-7 


,, 


- 123-8 


42-1 


,, 


30-92 




J 


32-0 


77-0 




271-8 


74-7 


55 


155-4 


78-9 


,, 


260-0 


54-9 


> 


280-6 


49-5 


J 5 


234-3 


62-1 


}1 


190-0 


36-9 


>f 



Observer. 



Sarrau. 



Andrews. 

Sarrau. 

Sajotschewsky. 



Hydrogen, 

Oxygen, 

Nitrogen, 

Carbonic Acid, 

Bisulphide of Carbon, 

Sulphurous Acid, 

Chloroform, 

Benzol, 

Alcohol, 

Ether, 



Conductivity. 

126. By the thermal conductivity of a substance 
at a given temperature is meant the value of k in the 
expression 



(1) 



where Q denotes the quantity of heat that flows, in time 
t, through a plate of the substance of thickness x, the area 
of each of the two opposite faces of the plate being A, 
and the temperatures of these faces being respectively 
Vj and -z> 2 , each differing but little from the given temper- 
ature. The lines of flow of heat are supposed to be 
normal to the faces, or, in other words, the isothermal 
surfaces within the plate are supposed to be parallel to 
the faces ; and the flow of heat is supposed to be steady, 
in other words, no part of the plate is to be gaining or 
losing heat on the whole. 



104 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

Briefly, and subject to these understandings, conduc- 
tivity may be denned as the quantity of heat that passes 
in unit time, through unit area of a plate whose thickness 
is unity, when its opposite faces differ in temperature by 
one degree. 

127. Dimensions of Conductivity. From equation (1) 
we have 

= _Q (2) 

v. 2 - v l At 

The dimensions of the factor are simply M. since 

v 2 -v l 

the unit of heat varies jointly as the unit of mass and 
the length of the degree. The dimensions of the factor 

are - ; hence the dimensions of k are =_ . This is 

on the supposition that the unit of heat is the heat 
required to raise unit mass of water one degree. In 
calculations relating to conductivity it is perhaps more 
usual to adopt as the unit of heat the heat required to 
raise unit volume of water one degree. The dimensions of 

O L" 

will then be L 3 , and the dimensions of k will be 7= 

v^-v 

These conclusions may be otherwise expressed by say- 

M 

ing that the dimensions of conductivity are ^ when the 

JL1 

thermal capacity of unit mass of water is taken as unity, 

T ^ 
and are when the capacity of unit volume of water is 

taken as unity. In the C.G.S. system the capacities of 
unit mass and unit volume of water are practically 
identical. 



ix.] HEAT. 105 

128. Let c denote the thermal capacity of unit volume 
of a substance through which heat is being conducted. 

Then - denotes a quantity whose value it is often neces- 

C 

sary to discuss in investigations relating to the transmis- 
sion of heat. We have, from equation (2), 



c v 2 v l At 

Q k 

where Q' denotes . Hence - would be the numerical 
c c 

value of the conductivity of the substance, if the unit of 
heat employed were the heat required to raise unit volume 
of the substance one degree. Professor Clerk Maxwell 

k 
proposed to call - the tJiermometric conductivity, as dis- 

tinguished from k the thermal or calorimetric conductivity. 
We prefer, in accordance with Sir Wm. Thomson's article, 

k 
" Heat," in the Encyclopaedia Britannica, to call - the 

diffusivity of the substance for heat, a name which is 

k 
based on the analogy of - to a coefficient of diffusion. 

C 

Coefficient of Diffusion. 

129. There is a close analogy between conduction and 
diffusion. Let x denote the distance between two 
parallel plane sections A and B to which the diffusion is 
perpendicular, and let these sections be maintained in 
constant states. Then, if we suppose one substance to be 
at rest, and another substance to be diffusing through it, 
the coefficient of diffusion K is defined by the equation 

y=K* ...... (i) 



106 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

where y denotes the thickness of a stratum of the mixture 
as it exists at B, which would be reduced to the state 
existing at A by the addition to it of the quantity which 
diffuses from A to B in the time t. 

When the thing diffused is heat, the states at A and B are 
the temperatures v 1 and v 2 , and y denotes the thickness of 
a stratum at the lower temperature which would be raised 
to the higher by the addition of as much heat as passes 
in the time t. This quantity of heat, for unit area, will be 

kt , 

-(%-%) 

JL 

which must therefore be equal to 
yc(v 2 - vj, 
whence we have 

k t 

y = ~ -- 

C X 

k 

The "thermometric conductivity" - may therefore be re- 

c 

garded as the coefficient of diffusion of heat. 

130. When we are dealing with the mutual inter- 
diffusion of two liquids, or of two gases contained in a 
closed vessel, subject in both cases to the law that the 
volume of a mixture of the two substances is the sum of 
the volumes of its components at the same pressure, the 
quantity of one of the substances which passes any section 
in one direction must be equal (in volume) to the quantity 
of the other which passes it in the opposite direction, 
since the total volume on either side of the section 
remains unaltered ; and a similar equality must hold for 
the quantities which pass across the interval between 
two sections, provided that the absorption in the interval 



ix.] HEAT. 107 

itself is negligible. Let x as before denote the distance 
between two parallel plane sections A and B to which the 
diffusion is perpendicular. Let the mixture at A consist 
of in parts by volume of the first substance to 1 - in of 
the second, and the mixture at B consist of n parts of 
the second to 1 - n of the first, m being greater than 1 - n, 
and therefore n greater than 1 - m. The first substance 
will then diffuse from A to B, and the second in equal 
quantity from B to A. Let each of these quantities be 
such as would form a stratum of thickness z (the vessel 
being supposed prismatic or cylindrical, and the sections 
considered being normal sections), then s will be propor- 
tional to 

m (\ n) m + n 1 

^t. that is to t. 

x x 

and the coefficient of interdiffusion K is defined by the 
equation 

_ 1 < ..... (2) 



The numerical quantity m + n - 1 may be regarded as 
measuring the difference of states of the two sections 
A and B. 

If y now denote the thickness of a stratum in the con- 
dition of B which would be reduced to the state existing 
at A by the abstraction of a thickness z of the second 
substance, and the addition of the same thickness of the 
first, we have (\n}y + z as the expression for the 
quantity of the first substance in the stratum after the 
operation. This is to be equal to my. Hence we have 



108 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

^ind substituting for z its value in (2) we have finally 

= K|, (4) 

which is of the same form as equation (1), y now denoting 
the thickness of a stratum of the mixture as it exists at B, 
which would be reduced to the state existing at A by the 
addition to it of the quantity of one substance which 
diffuses from A to B in the time t, and the removal from 
it of the quantity of the other substance which diffuses 
from B to A in the same time. 

131. The following values of K in terms of the centi- 
metre and second are given in Professor Clerk Maxwell's 
" Theory of Heat," 4th edition, p. 332, on the authority of 
Professor Loschmidt of Vienna. 

Coefficients of Interdiffusion of Gases. 

Carbonic Acid and Air, -1423 

,, ,, Hydrogen, -5614 

,, ,, Oxygen, -1409 

,, Marsh Gas, -1586 

,, ,, Carbonic Oxide, -1406 

,, ,, Nitrous Oxide, -0982 

Oxygen and Hydrogen, '7214 

,, ,, Carbonic Oxide, '1802 

Carbonic Oxide and Hydrogen, '6422 

Sulphurous Acide and Hydrogen, '4800 

k 

These may be compared with the value of - for air, 

c 

which, according to Professor J. Stefan of Vienna, is '256. 
The value of k for air, according to the same authority, 
is 5*58 x 10~ 5 , and is independent of the pressure. Pro- 
fessor Maxwell, by a different method, calculates its value 
at 5-4 x 10-1 



ix.] HEAT. 109 

Results of Experiments on Conductivity of Solids. 
132. Principal Forbes' results for the conductivity of 
iron (Stewart on Heat, p. 261, second edition) are ex- 
pressed in terms of the foot and minute, the cubic foot 
of water being the unit of thermal capacity. Hence the 
value of Forbes' unit of conductivity, when referred to 

C.G.S., is pjr , or 15 '48; and his results must be 

multiplied by 15 '48 to reduce them to the C.G.S. scale. 
His observations were made on two square bars ; the side 
of the one being 1^ inch, and of the other an inch. The 
results when reduced to C.G.S. units are as follows : 



H-inch bar. 1-inch bar. 

......... -207 ......... '1536 

25 ......... -1912 ......... -1460 

50 ........ -1771 ......... -1399 

75 ......... -1656 ......... -1339 

100 ......... -1567 ......... '1293 

125 ......... -1496 ........ -1259 

150 ......... -1446 ......... -1231 

175 ......... -1399 ......... -1206 

200 ......... -1356 ......... -1183 

225 ......... -1317 ......... -1160 

250 ......... -1279 ......... -1140 

275 ......... -1240 ......... -1121 

133. Neumann's results ("Ann. de. Chim." vol. Ixvi. p. 
185) must be multiplied by -000848 to reduce them to 
our scale. They then become as follows : 

Copper, .............................. 1-108 

Brass, ................................. -302 

Zinc, ................................... -307 

Iron, ................................... -164 

German Silver, ..................... -109 

Ice, ......................... -0057 



110 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

In the same paper he gives for the following substances 

k k 
the values of i or - ; that is, the quotient of conductivity 

by the thermal capacity of unit volume. These require 
the same reducing factor as the values of k, and when 
reduced to our scale are as follows : 

Values of 1 

Coal, -00116 

Melted Sulphur, '00142 

Ice, -0114 

Snow, -00356 

Frozen Mould, -00916 

Sandy Loam, -0136 

Granite (coarse), '0109 

Serpentine, "00594 

134. Sir W. Thomson's results, deduced from observa- 
tions of underground thermometers at three stations at 
Edinburgh ("Trans. R. S. E.," 1860, p. 426), are given in 
terms of the foot and second, the thermal capacity of a 
cubic foot of water being unity, and must be multiplied 
by (30 -48) 2 or 929 to reduce them to our scale. The 
following are the reduced results : 

k, or k 

Conductivity. " c " 

Trap-rock of Calton Hill, -00415 '00786 

Sand of experimental garden, '00262 , '00872 

Sandstone of Craigleith Quarry, '01068 '0231 1 

My own result for the value of from the Greenwich 

C 

underground thermometers ("Greenwich Observations," 
1860) is in terms of the French foot and the year. As 
a French foot is 32'5 centims., and a year is 31557000 
seconds the reducing factor is (32'5) 2 -j- 31557000; that 
is, 3-347 x 10~ 5 . The result is fc 

c 
Gravel of Greenwich Observatory Hill, '01249 



ix.] HEAT. Ill 

Professors Ayrton and Perry (" Phil. Mag.," April, 
1878) determined the conductivity of a Japanese building 
stone (porphyritic trachyte) to be '0059. 

1 35. Angstrom, in " Pogg. Ann.," vols. cxiv. (1861) and 
cxviii. (1863), employs as units the centimetre and the 
minute ; hence his results must be divided by 60. These 
results, as given at p. 294 of his second paper, will then 
stand as follows : 

Value of -'. 
Copper, first specimen, ........ 1 "216 (1 - "00214 t) 

,, second specimen, ...... 1-163 (1 - '001519 t] 

Iron, ................................. -224 (1 - '002874 t) 

He adopts for c the values 

84476 for copper ; '88620 for iron, 
and thus deduces the following values of k : 

Conductivity. 
Copper, first specimen, ......... T027 (1 - '00214 t) 

second specimen, ...... '983 (1 - '001519 t) 

Iron, .................................. -199 (1 - '002874 t) 

136. A Committee, consisting of Professors Herschel 
and Lebour, and Mr. J. F. Dunn, appointed by the British 
Association to determine the thermal conductivities of 
certain rocks, have obtained results from which the 
following selection has been made by Professor Herschel : 

Substance. 



Iron pyrites, more than ........... '01 more than '0170 

Rock salt, rough crystal, .......... '0113 '0288 

Fluorspar, rough crystal, ......... '00963 '0156 

Quartz, opaque crystal, and 

quartzites, ......................... -0080 to '0092 '0175 to '0190 

Silicious sandstones (slightly wet), '00641 to '00854 '0130 to '0230 



112 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

Suhstanrp Conductivity in k 

C.G.S. Units. ~ c - 

Galena, rough crystal, inter- 
spersed with quartz, "00705 '0171 

Sandstone and hard grit, dry, ... -00545 to '00565 '0120 
Sandstone and hard grit, thor- 
oughly wet, -00590 to -00610 -0100 

Micaceous flagstone, along the 

cleavage, -00632 '0116 

Micaceous flagstone, across cleav- 
age, -00441 -0087 

Slate, along cleavage, '00550 to '00650 '0102 

Do. , across cleavage, -003 1 5 to '00360 '0057 

Granite, various specimens, about '00510 to '00550 '0100 to '0120 
Marbles, limestone, calcite, and 

compact dolomite, '00470 to '00560 '0085 to '0095 

Red Serpentine (Cornwall), '00441 '0065 

Caen stone (building limestone), -00433 '0089 

Whinstone, trap rock, and mica 

schist, '00280 to '00480 '0055 to '0095 

Clay slate (Devonshire), '00272 '0053 

Tough clay (sun-dried), '00223 '0048 

Do., soft (with one-fourth 

of its weight of water), 00310 '0035 

Chalk, '00200 to '00330 '0046 to '0059 

Calcareous sandstone (firestone), '00211 '0049 

Plate-glass German and English, '00198 to '00234 '00395 to ? 

German glass toughened, '00185 '00395 

Heavy spar, opaque rough crystal, '00177 

Fire-brick, '00174 '0053 

Fine red brick, '00147 '0044 

Fine plaster of Paris, dry plate, '00120 ' 006 \about 

Do., thoroughly wet, '00160 '0025 / 

White sand, dry, '00093 '0032 

Do. , saturated with 

water, about '00700 '0120 about 

House coal and cannel coal, '00057 to -00113 '0012 to '002T 

Pumice stone, '00055 



ix.] HEAT. 113 

137. Peclet in " Annales de Chimie," ser. 4, torn. ii. p. 
114 [1841], employs as the unit of conductivity the trans- 
mission, in one second, through a plate a metre square 
and a millimetre thick, of as much heat as will raise a 
cubic decimetre (strictly a kilogramme) of water one 
degree. Formula (2) shows that the value of this conduc- 
tivity in the C.G.S. system, is 

1000 T \j ...-. 1 

^ liooo > thatls ' Too' 

His results must accordingly be divided by 100. 

The same author published in 1853 a greatly extended 
series of observations, in a work entitled "Nouveaux 
documents relatifs aux chauffage et a la ventilation." In 
this series the conductivity which is adopted as unity is 
the transmission, in one hour, through a plate a metre 
square and a metre thick, of as much heat as will raise a 
kilogramme of water one degree. This conductivity, in 
C.G.S. units is 

1000 100 _J_ m , . 1 

1 '10000 '3600' S '360' 

The results must therefore be divided by 360. Those of 
them which refer to metals appear to be much too small. 
The following are for badly conducting substances : 

Density. Conductivity. 
Fir, across fibres, -48 '00026 

,, along fibres, -48 '00047 

Walnut, across fibres, '00029 

,, alongfibres, '00048 

Oak, across fibres, '00059 

Cork, -22 '00029 

Caoutchouc, '00041 

Guttapercha, '00048 

Starchpaste, 1-017 '00118 

Glass, 2'44 '0021 

H 



114 



UNITS AND PHYSICAL CONSTANTS. [CHAP. 



Density. Conductivity. 

Glass, 2-55 '0024 

Sand, quartz, 1-47 "00075 

Brick, pounded, coarse-grained, . 1 '0 "00039 
passed through { , . 
silk sieve,.... I 1 ' 6 
Fine brick dust, obtained by decan- \ , .__ 

tation, J L 

Chalk, powdered, slightly damp, '92 '00030 

,, washed and dried, "85 '00024 

washed, dried, and \ , . ft9 , nft9Q 

compressed, J 1 

Potato-starch, '71 '00027 

Wood-ashes, -45 '00018 

Mahogany sawdust, '31 '00018 

Wood charcoal, ordinary,powdered, '49 '00022 
Bakers' breeze, in powder, passed ^ . - 

throughsilk sieve, .../ 2o 

Ordinary wood charcoal in powder, ) ,, -nnnoor 

passed through silk sieve, \ ' 

Coke, powdered, '77 "00044 

Iron filings, 2"05 '00044 

Binoxide of man ganese, 1 '46 "00045 

Woolly Substances. 

Cotton Wool, of all densities, "0001 1 1 

Cotton swansdown (molleton de ) .nnm 1 1 

coton), of all densities, / "' 

Calico, new, of all densities, "000139 

Wool, carded, of all densities, "000122 

Woollen swansdown (molleton de \ nnnnfi'7 

laine) of all densities, / "" 

Eider-down, "000108 

. Hempen cloth, new, "54 -000144 

old, "58 "000119 

Writing-paper, white, '85 "0001 19 

Grey paper, unsized, "48 '000094 

138. In Professor George Forbes's paper on conductivity 
(" Proc. K. S. E.," February, 1873) the units are the centim. 
the minute ; hence his results must be divided by 60. 



IX.] 



HEAT. 



115 



In a letter dated March 4, 1884, to the author of this 
work, Professor Forbes remarks that the mean tempera- 
ture of the substances in these experiments was - 10, and 
expresses the opinion that bad conductors (such as most 
of these substances) conduct worse at low than at high 
temperatures an opinion which was suggested by the 
analogy of electrical insulators. His results reduced to 
C.G.S. are- 



Ice, along axis, '00223 

Ice, perpendicular to \ nfto1 o 

axis, J ' UUZ1 ^ 

Black marble, '00177 

White marble, "00115 

Slate, '00081 

Snow, -00072 

Cork, -000717 

Glass, -0005 

Pasteboard, -000453 

Carbon, -000405 

Roofing-felt, "000335 

Fir, parallel to fibre, -0003 
Fir, across fibre, and) 

along radius, / 

Boiler-cement, 

Paraffin, 

Sand, very fine, 



00011 
000 089 



000 088 

000162 

00014 

000131 



Kamptulikon, 

Vulcanized india- 
rubber, 

Horn, ' '000087 

Beeswax, 'OU0087 

Felt, -000087 

Vulcanite, '0000833 

Haircloth, "0000402 

Cotton-wool,divided, '000 0433 
pressed, '0000335 

Flannel, '0000355 

Coarse linen, '000 0298 

Quartz, along axis, '000922 
00124 
00057 
"00083 
Quartz, perpendicular) nn/in 

to axis, /' UU4tJ 

0044 



Sawdust, "000123 

Professor Forbes quotes a paper by M. Lucien De la 
Rive ("Soc. de Ph. et d'Hist. Nat. de Geneve," 1864) in 
which the following result is obtained for ice, 
Ice, -00230. 

M. De la Rive's experiments are described in " Annales 
de Chimie," ser. 4, torn. i. pp. 504-6. 

139. Dr. Robert Weber ("Bulletin, Soc. Sciences Nat. 
de Neufchatel," 188), has found the following conductivi- 
ties and surface emissivities for five specimens of rock 
from the St. Gothard tunnel : 



116 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

SPECIMEN No. 168. Micaceous Gneiss. 

Conductivity, '000917 + '0000044? 

Emissivity, '000185 + '0000023? 

Specific Heat, '1778 4- '00042? 

SPECIMEN No. 114. Mica Schist. 

Conductivity, '000733 + '000010? 

Emissivity, '000207 + '0000016? 

Specific Heat, '18000 + '00044? 

SPECIMEN No. 124. Eurite. 

Conductivity, '000862 + '00016)5 

Emissivity, '000249 + "000 000 09* 

Specific Heat, '1682 + "0006* 

SPECIMEN No. 140. Gneiss. 

Conductivity, '0014 + '000 003? 

Emissivity, '00026 + '0000008? 

Specific Heat, '1463 +'0009? 

SPECIMEN No. 146. Micaceous Schist. 

Conductivity, '000952 + '000009? 

Emissivity, '000168 + '0000023? 

Specific Heat, '1697 +'0006? 

Conductivity of Liquids. 

140. The conductivity of water, according to experi- 
ments by Mr. J. T. Bottomley ("Phil. Trans." 1881, 
April 3), is '002, which is nearly the same as the con- 
ductivity of ice. (See 138.) 

141. H. F. Weber ("Sitz. kon. Preuss. Akad." 1885), 
has made the following determinations of conductivities 
of liquids at temperatures of from 9 to 15 C. He em- 
ploys the centimetre, the gramme, and the minute as 
units : we have accordingly divided the original numbers 
by 60 to reduce to C.G.S. 



IX.] 



HEAT. 



117 



Water, 


Conduc- 
tivity. 
00136 
000408 
000670 
000303 

000495 
000423 
000373 
000340 
000328 

000648 
000472 
000390 
000360 
000340 
000325 
000312 
000298 

000385 
000378 
000348 
000357 
000327 
000335 
000318 
000315 
000307 


Amyl Acetate, 
Chloro Benzol, . . 


Conduc- 
tivity. 
000302 

000302 
000288 
000252 
000283 
000278 
000284 

000265 
000247 
000257 
000278 
000237 

000222 
000220 
000208 
000203 

000333 
000307 
000272 
000260 

000765 
000343 
000382 
000328 


Aniline, 


Glycerine 


Ether, 


Chloroform, 
Chloro Carbon, 


Methyl Alcohol, 
Ethyl Alcohol, 
Propyl Alcohol, 
Butyl Alcohol, 
Amyl Alcohol, 

Ameisen Acid, 
Acetic Acid, 
Propion Acid, 
Butyric Acid, 
Isobutyric Acid, 
Valerian Acid, 
Isovalerian Acid, 


Propyl Chloride, 
Isobutyl Chloride, .... 
Amyl Chloride, . . 


Bromo Benzol, 
Ethyl Bromide, 
Propyl Bromide, 
Isobutyl Bromide,.... 
Amyl Bromide 

Ethyl Iodide, 


Propyl Iodide, 
Isobutyl Iodide, 
Amyl Iodide, 


Isocaproii Acid, 

Methyl Acetate, 
Ethyl Formiate, 
Ethyl Acetate, 


Benzol, . . 


Toluol, 


Cymol, 


Propyl Formiate, 
Propyl Acetate, 
Methyl Butyrate, 
Ethyl Butyrate, 
Methyl Valerate, 
Ethyl Valerate.... 


Oil of Turpentine, .... 

Sulphuric Acid, 
Bisulphide of Carbon, 
Oil of Mustard, 
Ethvl Suluhide. . . 



In the original paper these numbers are compared with 
the thermal capacities of the liquids per unit of volume, 
and with the calculated mean distances between their 
molecules. It is found that conductivity, multiplied by 
mean distance, divided by capacity, is a nearly constant 
quantity for the members of any one of the above groups. 
Comparing one group with another, its most widely dif- 
ferent values are represented by 19 and 23, if we except 
the last group, for which its value is between 26 and 27. 
Emission and Surface Conduction. 

142. Mr. D. M'Farlane has published (" Proc. Roy. Soc." 



118 



UNITS AND PHYSICAL CONSTANTS. [CHAP. 



1872, p. 93) the results of experiments on the loss of heat 
from blackened and polished copper in air at atmospheric 
pressure. They need no reduction, the units employed 
being the centimetre, gramme, and second. The general 
result is expressed by the formulas 

x = -000238 + 3-06 x lO' 6 * - 2-6 x 1Q-V 
for a blackened sui-face, and 

x = -000168 + 1 -98 x 10-^-1-7 x !Q- 
for polished copper, x denoting the quantity of heat lost 
per second per square centim. of surface of the copper, 
per degree of difference between its temperature and that 
of the walls of the enclosure. These latter were blackened 
internally, and were kept at a nearly constant temperature 
of 14 C. The air within the enclosure was kept moist 
by a saucer of water. The greatest difference of tempera- 
ture employed in the experiments (in other words, the 
highest value of t) was 50 or 60 C. 

The following table contains the values of x calculated 
from the above formula?, for every fifth degree, within the 
limits of the experiments : 



Difference of 
Temperature. 


Value of x. 


Ratio. 


Polished Surface. 


Blackened Surface. 


5 


000178 


000252 


707 


10 


000186 


000266 '699 


15 


000193 


000279 '692 


20 


000201 


000289 '695 


25 


000207 


000298 '694 


30 


000212 


000306 '693 


35 


000217 


090313 -693 


40 


000220 


000319 '693 


45 


000223 


000323 


690 


50 


000225 


000326 


690 


55 


000226 


000328 


690 


60 -000226 


000328 '690 



IX.] 



HEAT. 



119 



143. Professor Tait has published (" Proc. R. S. E." 
1869-70, p. 207) observations by Mr. J. P. Nichol on the 
loss of heat from blackened and polished copper, in air, 
at three different pressures, the enclosure being blackened 
internally and surrounded by water at a temperature of 
approximately 8 C.* Professor Tait's units are the grain- 
degree for heat, the square inch for area, and the hour for 
time. The rate of loss per unit of area is 

heat emitted 
area x time 

The grain-degree is '0648 gramme-degree. 
The square inch is 6*4516 square centims. 
The hour is 3600 seconds. 
Hence Professor Tait's unit rate of emission is 

*0648 n.rrn 1 /->-<{ 



6-4516 x 3600 

of our units. Employing this reducing factor, Professor 
Tait's Table of Results will stand as follows : 

Pressure 1 '014 x 10 6 [760 millims. of mercury]. 

Bright. 

Temp. Cent. Loss per sq. cm. 

per second. 

63-8 -00987 

57-1 -00862 

50-5 -00736 

44-8 -00628 

40-5 -00562 

34-2 -00438 

29-6 -00378 

23-3 -00278 

18-6 . -00210 



Blackened. 

Temp. Cent. Less per sq. cm. 

per second. 

61-2 -01746 

50-2 -01360 

41-6 '01078 

34-4 "00860 

27-3 -00640 

20-5 -00455 



*This temperature is not stated in the " Proceedings," but has 
been communicated to me by Professor Tait. 



120 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

Pressure 1'36-x 10 5 [102 millims. of mercury]. 

62-5 -01298 I 67-8 '00492 

57'5 -01158 61-1 -00433 

53-2 -01048 55 -00383 



47-5 -00898 

43 -00791 

28-5 . -00490 



497 '00340 

44-9 -00302 

40-8 . -00268 



Pressure P33 x 10 4 [10 millims. of mercury]. 



62-5 


01182 


65 


00388 


57'5 
54-2 


... -01074 
... -01003 


60 
50 


-00355 
. ... -00286 


41-7 


. . . -00726 


40 


00219 


37-5 
34 


.. -00639 
00569 


30 
23-5 


-00157 
00124 


27-5 


... -00446 






24-2 . 


00391 







Mechanical Equivalent of Heat. 

144. The value originally deduced by Joule from his 
experiments on the stirring of water was 772 foot-pounds 
of work (at Manchester) for as much heat as raises a 
pound of water through 1 Fahr. This is 1389-6 foot- 
pounds for a pound of water raised 1 C., or 1389-6 foot- 
grammes for a gramme of water raised 1 C. As a foot 
is 30 '48 centims., and the value of g at Manchester is 
981-3, this is 1389-6 x 30-48x981-3 ergs per gramme- 
degree; that is, 4-156 x 10 7 ergs per gramme- degree. 

A later determination by Joule (" Brit. Assoc. Report," 
1867, pt. i. p. 522, or "Reprint of Reports on Electrical 
Standards," p. 186) is 25187 foot-grain-seconcl units of 
work per grain-degree Fahr. This is 45337 of the same 
units per grain-degree Centigrade, or 45337 foot-gramme- 
second units of work per gramme-degree Centigrade; 
that is to say, 

45337 x (30-48) 2 = 4-212 x 10 7 
ergs per gramme-degree Centigrade. 



ix.] HEAT. 121 

In view of the fact that the B. A. standard of electrical 
resistance employed in this determination is now known 
to be too small by about 1/3 per cent., and that the cur- 
rent energy converted into heat was accordingly under- 
estimated to this extent, the result ought now to be in- 
creased by 1'3 per cent., which will make it 
4-267 x 107. 

At the meeting of the Royal Society, January, 1878 
(" Proceedings," vol. xxvii. p. 38), an account was given by 
Joule of experiments recently made by him with a view 
to increase the accuracy of the results given in his former 
paper. ("Phil. Trans.," 1850.) His latest result from 
the thermal effects of the friction of water, as announced 
at this meeting, is, that taking the unit of heat as that 
which can raise a pound of water, weighed in vacuo, from 
60 to 61 of the mercurial Fahrenheit thermometer; its 
mechanical equivalent, reduced to the sea-level at the 
latitude of Greenwich, is 772-55 foot-pounds. 

To reduce this to water at C. we have to multiply 
by 1-00089,* giving 773-24 ft. Ibs., and to reduce to ergs 
per gramme-degree Centigrade we have to multiply by 

981-17 x 30-48 x-. 
5 

The product is 4-1624 x 10 7 . 

145. Some of the best determinations by various experi- 
menters are given (in gravitation measure) in the following 
list, extracted from "Watts' Dictionary of Chemistry," 
Supplement 1872, p. 687. The value 429 -3 in this list 
corresponds to 4*214 x 10 7 ergs : 

* This factor is found by giving t the value 15 '8 (since the tem- 
perature GO'S Fahr. is 15 '8 Cent.) in formula (3) of art. 101. 



122 



UNITS AND PHYSICAL CONSTANTS. [CHAP. 



Violle,... 



Hirn, 432 Friction of water and brass. 

,, 433 Escape of water under pressure. 

..... 441-6 Specific heats of air. 

,, 425-2 Crushing of lead. 

Joule 429-3 ^ Heat produced by an electric 

( current. 
435 -2 (copper).. . 
434*9 (aluminium) 

435-8 (tin) 

437-4 (lead) 

Regnault, 437 

We shall adopt 4'2 x 10 7 ergs as the equivalent of 
1 gramme-degree ; that is, employing J as usual to denote 
Joule's equivalent, we have 

J = 4-2 x 10 7 = 42 millions. 
146. Heat and Energy of Combination with Oxygen. 



Heat produced by induced cur- 
rents. 

Velocity of sound. 



1 gramme of 


Compound 
formed. 


Gramme- 
degrees of heat 
produced. 


Equivalent 
Energy, in 
ergs. 


Hydrogen, 
Carbon, 


H 2 

CO 2 


34000 A F 
8000 A F 


l-43x!0 12 
3'36x 10 11 


Sulphur, . 


SO 2 


2300 A F 


9'66x 10 10 


Phosphorus, 
Zinc, 


P 2 5 
ZnO 


5747 A 
1301 A 


2-41 x 10 11 
5-46x 10 10 


Iron, 


Fe 3 4 


1576 A 


6'62x 10 10 


Tin,. 


SnO' 


1233 A 


5-18 


Copper, 


CuO 


602 A 


2-53 


Carbonic oxide,.... 
]VIarsh-gas 


CO 2 
CO 2 and H' J 


2420 A 
13100 A F 


1 -02 x 10 11 
5'50 


Olefiant gas, . . 




11900 A F 


5-00 


Alcohol, 




6900 A F 


2-90 




" 







Combustion in Chlorine. 



Hydrogen, ; HC1 
Potassium, KC1 
Zinc, ZnCP 


i 23000 F T 
2655 A 
1529 A 


9-66xlO n 
1-12 
6-42xl0 10 


Iron Fe 2 Cl fi 


1745 A 


7-33 


Tin, . . SnCl 4 


1079 A 


4-53 


Copper CuCl 2 


961 A 


4'04 









ivl 1 1 MAT. 123 

The numbers in the last column are the products of the 
numbers in the preceding column by 42 millions. 

The authorities for these determinations are indicated 
by the initial letters A (Andrews), F (Favre and Silber- 
mann), T (Thomsen). Where two initial letters are 
given, the number adopted is intermediate between those 
obtained by the two experimenters. 

147. Difference between the two specific heats of a gas. 
Let s 1 denote the specific heat of a given gas at con- 
stant pressure, 

s 2 the specific heat at constant volume, 
a the coefficient of expansion per degree. 
v the volume of 1 gramme of the gas in cubic centim. 

at pressure/* dynes per square centim. 
When a gramme of the gas is raised from to 1 at 
the constant pressure^;, the heat taken in is s v the increase 
of volume is av, and the work done against external resist- 
ance is avp (ergs). This work is the equivalent of the 
difference between s 1 and s 2 ; that is, we have 

s x ,s- 2 = _ JL, where J = 4'2 x 20 7 . 
J 

For dry air at the value of vp is 7*838 x 10 8 , and a 
is -003665. Hence we find s l - s 2 = -0684. The value of 
s 15 according to Eegnault, is -23 7f>. Hence the value of 
s z is -1691. 

The value of 1 ~ S -, or ^ . for dry air at and a 
v J 

megadyne per square centim. is 

*i -*2-' 0684 

" v 7S3-8 



1 24 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

and this is also the value of - 1 2 for any other gas (at 

the same temperature and pressure) which has the same 
coefficient of expansion. 

148. Change of freezing point due to change of pressure. 
Let the volume of the substance in the liquid state be 
to its volume in the solid state of 1 to 1 + e. 

When unit volume in the liquid state solidifies under 
pressure P + p, the work done by the substance is the 
product of P +2> by the increase of volume e } and is there- 
fore ~Pe +pe. 

If it afterwards liquefies under pressure P, the work 
done against the resistance of the substance is Pe ; and if 
the pressure be now increased to P + p, the substance will 
be in the same state as at first. 

Let T be the freezing temperature at pressure P, 

T + t the freezing temperature at pressure P +p, 
I the latent heat of liquefaction, 
d the density of the liquid. 

Then d is the mass of the substance, and Id is the heat 
taken in at the temperature of melting T. Hence, by 
thermodynamic principles, the heat converted into mechani- 
cal effect in the cycle of operations is 



T + 273 ' 

But the mechanical effect is pe. Hence we have 
t 77 pe 



p 3 Id 



ix.] HEAT. 125 

- - is the lowering of the freezing-point for an additional 

pressure of a dyne per square centim.; and x 10 will 

be the lowering of the freezing point for each addi- 
tional atmosphere of 10 6 dynes per square centim. 
For water we have 

e = 'OS7, Z= 79-25, T = 0, d = l, 



Formula (3) shows that is opposite in sign to e. 

Hence the freezing point will be raised by pressure if the 
substance contracts in solidifying. 

149. Change of temperature produced by adiabatic com- 
pression of a fluid ; that is, by compression under such 
circumstances that no heat enters or leaves the fluid. 

Let a cubic centim. of fluid at the initial temperature 
t C. and pressure p dynes per square centim. be put 
through the cycle of operations represented by the annexed 
"indicator diagram," ABCD, where horizontal distance 
from left to right denotes increase of volume and perpen- 
dicular distance upwards increase of pressure. 

In AD let the pressure be constant 
and equal to p. 

In BC let the pressure be constant 
and equal to p + TT, TT being small. 

Let AB and CD be adiabatics, so near 
together that AD and BC are very small 
compared with the altitude of the figure 
which is TT. 




126 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

The figure will be ultimately a parallelogram, so that 
the changes of volume AD and BC will be equal ; let their 
common value be called edt, e denoting the expansion per 
degree at constant pressure; dt will therefore be the 
difference of temperature between A and D, or between 
B and C. We suppose this difference to be very small 
compared with the difference of temperature between A 
and B or between C and D. 

The cycle is reversible; let it be performed in the direc- 
tion ABCD. Then heat is taken in as the substance 
expands from B to C, and given out as it contracts from 
Dto A. 

The work done by the substance in the cycle is equal 
to the area of the parallelogram, which, being the product 
of the base edt by the height TT, is Tredt. The heat given 
out in DA is Cdt, C denoting the thermal capacity of a 
cubic centim. of the substance at constant pressure; 

hence the " efficiency " is JT , and this, by the rules of 

JC 

Thermodynamics, must be equal to -j - , where r de- 

i i o ~\~ t 

notes the increase of temperature from A to B. Put T 
for the absolute temperature 273 + 2, then we have 



where r is the increase of temperature produced by the 
increase TT of pressure. 

150. Resilience as affected by heat of compression. 
The expansion due to the increase of temperature r, 

above calculated, is re ; that is, - ~~ ; and the ratio of 

J C 



ix.] HEAT. 127 



this expansion to the contraction , which would be pro- 

Jfi 

duced at constant temperature (E denoting the resilience 



of volume at constant temperature), is =~- : 1. Putting 

JO 



m for - , the resilience for adiabatic compression will be 
JC 

TT 

; or, if m is small, E (1 + m) \ and this value is to 



l-m 

be used instead of E in calculating the change of volume 
due to sudden compression. 

The same formula expresses the value of Young's 
modulus of resilience, for sudden extension or compression 
of a solid in one direction, E now denoting the value of 
the modulus at constant temperature. 

Examples. 

For compression of water between 10 and 11 we have 
E = 2-1 x 10 10 , T = 283, e = -000 092, C - 1 ; 

hence 



For longitudinal extension of iron at 10 we have 
E = l-96xl0 12 , T-283, e=-0000122, C = '109x7 



hence = -00234. 

JG 

Thus the heat of compression increases the volume- 
resilience of water at this temperature by about -| per 
cent., and the longitudinal resilience of iron by about 
J per cent. 



128 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

For dry air at and a megadyne per square centim., 
we have 

E = 10 6 , T = 273, e - _* C - -2375 x -001276, 
27o 

= 1-404. 



1 -m 

151. Expansions of Volumes per degree Cent, (abridged 
from Watts' " Dictionary of Chemistry" Article Heat, pp. 
67, 68, 71). 

Glass, ........................... -00002 to "000 03 

Iron .............................. -000035 '000044 

Copper, ......................... -000052 ,, -000057 

Platinum, ..................... '000026 ,, '000029 

Lead, ............................ -000084 -000089 

Tin, ............................. -000058 '000069 

Zinc, ............ . ................ -000087 '000090 

Gold, ............................ -000044 ,, -000047 

Brass, ........................... -000053 -000056 

Silver, ........................ -000057 '000064 

Steel, ............................ -000032 -000042 

Cast Iron, ............... about '000033 

These results are partly from direct observation, and 
partly calculated from observed linear expansion. 

Expansion of Mercury, according to Regnault (Watts' 
"Dictionary" p. 56). 

Te mP . = , Volume at, 

6 ............... 1 '000000 ............... '00017905 

10 ............... 1-001792 .............. -00017950 

20 ............... 1-003590 ............... -00018001 

30 ............... 1-005393 ............... -00018051 

50 ......... ...... 1-009013 ............... -00018152 

70 ............... 1-012655 ............... -00018253 

100 ............... 1-018153 ............... -00018405 

The temperatures are by air-thermometer. 



ix.] , HEAT. 129 

The formula adopted by the Bureau International dcx 
Poids et Mesures for the volume at t C. (derived from 
Regnault's results) is 

1 + -000181792*+ -000 000 000 175* 2 

+ 000000000035116^. 

Expansion of Alcohol and Ether, according to Kopp 
(Watts' "Dictionary," p. 62). 

Volume- 
Temp. Alcohol. Ether. 

6 roooo roooo 

10 1-0105 1-0152 

20 1-0213 1-0312 

30 1-0324 1-0483 

40 . 1-0440 , 1-0667 



152. Collected Data for Dry Air. 

Expansion from to 100 at const, pressure, as 1 to 1*367 

or as 273 to 373 

Specific heat at constant pressure, '2375 

,, at constant volume, -1691 

Pressure-height at C., about 7 '99 x 10 5 cm., 

or about 26210 ft. 

Standard barometric column, 76 cm. = 29 '922 inches. 

Standard pressure, 1033 '3 gm. per sq. cm. 

or 14*7 Ibs. per sq. inch. 

or 2117 Ibs. ,, foot. 

or 1 '0136 x 10 6 dynes per sq. cm. 

Standard density, at C., "001293 gm. per cub. cm. 

or -0807 Ibs. per cub. foot. 

Standard bulkiness, 773 '3 cub . cm. per gm. 

or 12-39 cub. ft. per Ib. 
I 



130 UNITS AND PHYSICAL CONSTANTS. [CHAP. ix. 

Dry and Moist Ait'. 

Mass of 1 Cubic Metre in Grammes. 



Temp. C. 
() 


Dry Air. 

1293-1 


Saturated Air. 
1290*2 


Vapour at 
Saturation. 

4-9 


10 


... 1247-3 


. . 1241-7 


9-4 


20 
30 . 


... 1204-6 .... 
1164-8 


1194-3 
1146*8 


17-1 
. ... 30-0 


40 . 


1127-6 


1097-2 


50-7 



If A denote the density of dry air and W that of vapour at 
ituration, the den: 
exactly A - -608 W. 



o 

saturation, the density of saturated air is A - - W, or more 

5 



131 



CHAPTER X. 
MAGNETISM. 

153. THE unit magnetic pole, or the pole of unit strength, 
is that which repels an equal pole at unit distance with 
unit force. In the C.G.S. system it is the pole which 
repels an equal pole, at the distance of 1 centimetre, with 
a force of 1 dyne. 

If P denote the strength of a pole, it will repel an equal 

P 2 

pole at the distance L with the force . Hence we have 

the dimensional equations 

P 2 L~ 2 = force - MLT~ 2 , P 2 = ML 3 T' 2 , P = M^TT 1 ; 
that is, the dimensions of a pole (or the dimensions of 

strength of pole) are M^L^T' 1 . 

154. The work required to move a pole P from one 
point to another is the product of P by the difference of 
the magnetic potentials of the two points. Hence the 
dimensions of magnetic potential are 



>- - ML 2 T- 2 . M ~ *L ~ T 

155. The intensity of a magnetic field is the force which 
a unit pole will experience when placed in it. Denoting 



132 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

this intensity by I, the force on a pole P will be IP. 
Hence 



IP = force = MLT- J , 1 = MLT' 3 . MLT 

that is, the dimensions of 'field-intensity are M*L"*T~ 1 . 

156. The moment of a magnet is tbe product of the 
strength of either of its poles by the distance between 

them. Its dimensions are therefore LP; that is, M^UT' 1 . 
Or, more rigorously, the moment of a magnet is a 
quantity which, when multiplied by the intensity of a 
uniform field, gives the couple which the magnet ex- 
periences when held with its axis perpendicular to the 
lines of force in this field. It is therefore the quotient of 

a couple ML 2 T~ 2 by a field-intensity M^L'^T" 1 ; that 
is, it is M^L'T' 1 as before. 

157. If different portions be cut from a uniformly mag- 
netized substance, their moments will be simply as their 
volumes. Hence the intensity of magnetization of a uni- 
formly magnetized body is defined as the quotient of its 
moment by its volume. But we have 



moment 
volume 



_ 1 ^ L _ 3 = M 4 L -4 T -i p 



Hence intensity of magnetization has tbe same dimensions 
as intensity of 'field. When a magnetic substance (whether 
paramagnetic or diamagnetic) is placed in a magnetic 
field, it is magnetized by induction, and the ratio of the 
intensity of the magnetization thus produced to the 
intensity of the field is called the " coefficient of magnetic 



x.l MAGNETISM. 133 

induction," or " coefficient of induced magnetization," or 
the "magnetic susceptibility" of the substance. For 
paramagnetic substances (such as iron, nickel, and cobalt) 
this coefficient is positive ; for diamagnetic substances 
(such as bismuth), it is negative ; that is to say, the induced 
polarity is reversed, end for end, as compared with that 
of a paramagnetic substance placed in the same field. 

158. It has generally been stated that "magnetic sus- 
ceptibilty " is nearly independent of the intensity of the 
h'eld so long as this intensity is much less than is required 
for saturation. But R. Shida found ("Proc. Roy. Soc.," 
Nov., 1882), in the softest iron wire, a, very rapid varia- 
tion of susceptibility at low intensities. Under the 
influence of the earth's vertical force at Glasgow, *545, 
the susceptibility had the very large value 734 when the 
wire was stretched by a weight, and 335 when the weight 
was off. 

Under a magnetizing force 2 -35, the susceptibilities, 
with and without the weight, were 235 and 154. 

Saturation was obtained with a magnetizing force of 
80*7, which produced magnetizations 1390 and 1430, the 
susceptibilities being therefore 17*1 and 17 '6. 

With pianoforte wire (steel), the susceptibilities were 
67'5 and 69'3 under the earth's vertical force, and 13*2 
when saturation was just attained, with a magnetizing 
force of 107 '5. The magnetization at saturation was 
1420, being about the same as for soft iron wire. 

With a square bar of soft iron nearly 1 centim. square, 
the susceptibility diminished from 19, under a magnetizing 
force of 18 -2, to 7'6, under a magnetizing force of 189, 
which just produced saturation. 



134 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

Examples. 

1. To find the multiplier for reducing magnetic in- 
tensities from the foot-grain-second system to the C.G.S. 

system. 



The dimensions of the unit of intensity are 
In the present case we have M = -0648, L = 30'48, T = 1, 
since a grain is -0648 gramme, and a foot is 30 '48 centini. 



Hence M^L^T' 1 = ./^|= '04611 ; that is, the foot- 
\ olr4o 

grain-second unit of intensity is denoted by the number 
0461 1 in the C.G.S. system. This number is accordingly 
the required multiplier. 

2. To find the multiplier for reducing intensities from 
the millimetre-milligramme-second system to the C.G.S. 
system, we have 

M - T, T - 1 ~M^T,~^T~ l - / - 

" 1000' "16' ~V 1000" 10' 

Hence - - is the required multiplier. 

3. Gauss states (Taylor's " Scientific Memoirs," vol. ii. 
p. 225) that the magnetic moment of a steel bar-magnet, 
of one pound weight, was found by him to be 100S77000 
millimetre-milligramme-second units. Find its moment 
in C.G.S. units. 

Here the value of the unit moment employed is, in 

terms of C.G.S. units, M^T' 1 , where M is 10~ 3 , L is 

10- 1 , arid T is 1 ; that is, its value is 10~*. 10~ = 10~ 4 . 
Hence the moment of the bar is 10087*7 C.G.S. units. 



x.] MAGNETISM. 135 

4. Find the mean intensity of magnetization of the bar, 
assuming its specific gravity to be 7*85, and assuming that 
the pound mentioned in the question is the pound avoir- 
dupois of 453*6 grammes. 

Its mass in grammes, divided by its density, will be 
its volume in cubic centimetres \ hence we have 
453-6 



In! 

K ' 



7-85 



= 5778 = volume of bar. 



.. ,. moment 10088 ,, 

ntensity of magnetization = = = 174-0. 

volume 57-78 



5. Kohlrausch states (" Physical Measurements," p. 195, 
English edition) that the maximum of permanent mag- 
netism which very thin rods can retain is about 1000 
millimetre-milligramme-second units of moment for each 
milligramme of steel. Find the corresponding moment 
per gramme in C.G-.S. units, and the corresponding in- 
tensity of magnetization. 

For the moment of a milligramme we have 1000 
x 10-* =10-!. 

For the volume of a milligramme we have (7-85)" 1 
x 10~ 3 , taking 7 '85 as the density of steel. 

Hence the moment per gramme is 10" 1 x 10 3 = 100, and 
the intensity of magnetization is 100 x 7 "85 = 785. 

6. The maximum intensity of magnetization for speci- 
mens of iron, steel, nickel, and cobalt has been deter- 
mined by Professor Rowland ("Phil. Mag.," 1873, vol. 
xlvi. p. 157, and November, 1874) that is to say, the 
limit to which their intensities of magnetization would 
approach, if they were employed as the cores of electro- 
magnets, and the strength of current and number of con- 
volutions of the coil were indefinitely increased. Professor 



136 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

Rowland's fundamental units are the metre, gramme, and 
second ; hence his unit of intensity is of the C.G.S. 
unit. His values, reduced to C.G.S. units, are 

At 12C. At 220C. 

Iron and Steel, ......................... 1390 1360 

Nickel, .................................... 494 380 

Cobalt, .................................... 800 (?) 

7. Gauss states (loc. cit.) that the magnetic moment of 
the earth, in millimetre-milligramme-second measure, is 

3-3092 R 3 , 

R denoting the earth's radius in millimetres. Reduce 
this value to C.G.S. 

Since R 3 is of the dimensions of volume, the other 
factor, 3 '309 2, must be of the dimensions of intensity. 
Hence, employing the reducing factor 10" 1 above found, 
we have '33092 as the corresponding factor for C.G.S. 
measure \ and the moment of the earth will be 

33092 R 3 , 

R denoting the earth's radius in centimetres that is 
6-37 x 10 8 . 
We have 

33092 x (6-37 x 10 8 ) 3 - 8-55 x 10 25 
for the eartlis magnetic moment in C.G.S. units. 

8. From the above result, deduce the intensity of mag- 
netization of the earth regarded as a uniformly magnetized 
body. 

We have 

. , ., moment 8*55 x 10 25 n70r . 

intensity = _ ^7^7 = ' 790 ' 

volume 1-083 x 2 



This is about - --- of the intensity of magnetization of 



.\.] MAGNETISM. 137 

Gauss's pound magnet ; so that 2*2 cubic decimetres of 
earth would be equivalent to 1 cubic centiin. of strongly 
magnetized steel, if the observed effects of terrestrial mag- 
netism were due to uniform magnetization of the earth's 
substance. 

9. Gauss, in his papers on terrestrial magnetism, em- 
ploys two different units of intensity, and makes mention 
of a third as " the unit in common use." The relation 
between them is pointed out in the passage above referred 
to. The total intensity at Gottingen, for the 19th of 
July, 1834, was 4'7414 when expressed in terms of one 
of these units the millimetre-milligramme-second unit ; 
was 1357 when expressed in terms of the other unit em- 
ployed by Gauss, and 1'357 in terms of the "unit in 
common use." In C.G.S. measure it would be '47414. 

159. A first approximation to the distribution of mag- 
netic force over the earth's surface is obtained by assuming 
the earth to be uniformly magnetized, or, what is mathe- 
matically equivalent to this, by assuming the observed 
effects to be due to a small magnet at the earth's centre. 
The moment of the earth on the former supposition, or 
the moment of the small magnet on the latter, must be 

33092 KS, 

K denoting the earth's radius in centims. The magnetic 
poles, on these suppositions, must be placed at 
77 50' north lat., 296 29' east long., 
and at 77 50' south lat., 116 29' east long. 

The intensity of the horizontal component of terrestrial 
magnetism, at a place distant A from either of these 
poles, will be 

33092 sin A ; 



138 UNITS AND PHYSICAL CONSTANTS. [CHAP, 

the intensity of the vertical component will be 

66184 cos A; 

and the tangent of the dip will be 
2 cotan A. 

The magnetic potential, on the same supposition, will 
be 

33092 cos A, 
r 2 

r being variable. (See Maxwell, " Electricity and Mag- 
netism," vol. ii. p. 8.) Gauss's approximate expression 
for the potential and intensity at an arbitrary point on the 
earth's surface consists of four successive approximations, 
of which this is the first. 

160. According to " Airy on Magnetism," the place of 
greatest horizontal intensity is in lat. long. 259 E., 
where the value is '3733 ; the place of greatest total in- 
tensity is in South Victoria, about 70 S., 160 E., where 
its value is '7898, and the place of least total intensity is 
near St. Helena, in lat. 16 S., long. 355 E., where its 
value is -2828. 

161. The following mean values of the magnetic ele- 
ments at Greenwich have been kindly furnished by the 
Astronomer Royal (Dec., 1885) : 

West Declination, 18 15' '0 - (t - 1883) x 7' '80. 

Horizontal Force, '1809 + (t- 1883) x -00018. 

Dip, 67 31'-8-(- 1883) x T39. 

Vertical Force, 0'4374 - (t- 1883) x "00007. 

= Horizontal force x tan. dip. 

Each of these formulae gives the mean of the entire 
year t. 

162. According to J. E. H. Gordon ("Phil. Trans.," 
1877, with correction in "Proc. Roy. Soc.," 1883, pp. 



x.j MAGNETISM. 139 

4, 5), the rotation of the plane of polarisation between two 
points, one centimetre apart, whose magnetic potentials 
(in C.G.S. measure) differ by unity, is (in circular mea- 
sure) 

1-52381 x 10-* 

in bisulphide of carbon, for the principal green thallium 
ray, and is 

2-248 x 10- 
in distilled water, for white light. 

Mr. Gordon infers from Becquerel's experiments ("Comp. 
Rend.," March 31, 1879) that it is about 

3x 10~ 9 
for coal gas. 

According to Lord Rayleigh (" Proc. Roy. Soc.," Dec. 
29, 1884), the rotation for sodium light in bisulphide of 
carbon at 18 C. is -04202 minute. This is 

1-22231 x 10- 5 
in circular measure. 



140 



CHAPTER XI, 
P]LECTRICITY. 

Electrostatics. 

163. IF q denote the numerical value of a quantity of 
electricity in electrostatic measure, the mutual force be- 
tween two equal quantities q at the mutual distance I will 

be $-. In the C.G.S. system the electrostatic unit of 

i~ 

electricity is accordingly that quantity which would repel 
an equal quantity at the distance of 1 centim. with a force 
of 1 dyne. 

Since the dimensions of force are , we have, as 

regards dimensions, 

q' 2 ml , ml 9 i,f - 1 

! = _-, whence f = ,q = mlt . 

164. The work done in raising a quantity of electricity 
q through a difference of potential v is qv. 

Hence we have 



In the C.G.S. system the unit difference of potential is 



CHAP, xi.] ELECTRICITY. HI 

that difference through which a unit of electricity must be 
raised that the work done may be 1 erg. 

Or, we may define potential as the quotient of quantity 
of electricity by distance. This gives 

v = m^ftt' 1 . l~ l = mh*t~\ as before. 

165. In the C.G.S. system the unit of potential is the 
potential due to unit quantity at the distance of 1 centim. 

The capacity of a conductor is the quotient of the 
quantity of electricity with which it is charged by the 
potential which this charge poduces in it. Hence we 
have 

capacity = ? = m^ftr 1 . m ~ fy ~^t = l. 

The same conclusion might have been deduced from 
the fact that the capacity of an isolated spherical con- 
ductor is equal (in numerical value) to its radius. The 
C.G.S. unit of capacity is the capacity of an isolated 
sphere of 1 centim. radius. 

166. The numerical value of a current (or the strength 
of a current) is the quantity of electricity that passes in 
unit time. 

Hence the dimensions of current are ^; that is, m*ftt~*. 

The C.G.S. unit of current is that current which con- 
veys the above denned unit of quantity in 1 second. 

167. The dimensions of resistance can be deduced from 
Ohm's law, which asserts that the resistance of a wire is 
the quotient of the difference of potential of its two ends, 
by the current which passes through it. Hence we have 

resistance = m^fir 1 . m ~fy~ V = T V 



142 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

Or, the resistance of a conductor is equal to the time 
required for the passage of a unit of electricity through it, 
when unit difference of potential is maintained between 
its ends. Hence 

time x potential 1 7 i -i T -4 7 -i 

resistance = - L__. _ = * . m 2 W" 1 . m 2 l ?t = I t. 
quantity 

168. As the force upon a quantity q of electricity, in a 
field of electrical force of intensity i, is iq, we have 



The quantity here denoted by i is commonly called the 
<c electrical force at a point." 

Electromagnetics. 

169. A current C (or a current of strength C) flowing 
along a circular arc, produces at the centre of the circle an 
intensity of magnetic field equal to C multiplied by length 
of arc divided by square of radius. Hence C divided by 
a length is equal to a field-intensity, or 

G = length x intensity = L . M^L " ^T" 1 = lAlV 1 . 

170. The quantity of electricity Q conveyed by a cur- 
rent is the product of the current by the time that it lasts. 

The dimensions of Q are therefore L*M 2 . 

171. The work done in urging a quantity Q through a 
circuit, by an electromotive force E, is EQ ; and the work 
done in urging a quantity Q through a conductor, by 
means of a difference of potential E between its ends, is 
EQ. Hence the dimensions of electromotive force, and 
also the dimensions of potential, are ML 2 T~ 2 . L " ^M ~ 2 , or 



ELECTRICITY. 



143 



172. The capacity of a conductor is the quotient of 
quantity of electricity by potential. Its dimensions are 
therefore 

M*L* . M ~ *L ~ T 2 ; that is, L^T 2 . 
-p 

173. Resistance is .-; its dimensions are therefore 



M*L*T- a . M ~ *L ~ *T ; that is, LT' 1 . 

174. The following table exhibits the dimensions of 
each electrical element in the two systems, together with 
their ratios : 





Dimensions in 
electrostatic 
system. 


Dimensions in 
electromagnetic 
system. 


Dimensions in E.S. 


Dimensions in E.M. 


Quantity, 


M*LT- 


M*L* 


LT- 1 


Current, 


M*L*T-> 


M*L*T-> 


LT- 1 


Capacity, 

Potential and ) 
electronic- > 
tive force, \ 


L 


L -1 T 2 


L 2 T- 2 


Resistance, 


L-T 


LT- 1 


L- 2 T 2 



175. The heat generated in time T by the passage of a 

current C through a wire of resistance E, (when no other 

p2"R r r 

work is done by the current in the wire) is gramme 

J 

degrees, J denoting 4 '2 x 10 7 ; and this is true whether C 
and R are expressed in electromagnetic or in electrostatic 
units. 



144 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

Ratios of the two sets of Electric Units. 

176. Let us consider any general system of units based 
on 

a unit of length equal to L centims., 
a unit of mass equal to M grammes, 
a unit of time equal to T seconds. 

Then we shall have the electrostatic unit of quantity 
equal to 

M^L^T' 1 C.G.S. electrostatic units of quantity, 
and the electromagnetic unit of quantity equal to 

M 5 L 2 C.G.S. electromagnetic units of quantity. 

It is possible so to select L and T that the electrostatic 
unit of quantity shall be equal to the electromagnetic 

unit. We shall then have (dividing out by M^lJ) 

LT- 1 C.G.S. electrostatic units 

= 1 C.G.S. electromagnetic unit; 
or the ratio of the C.G.S. electromagnetic unit to the 

C.G.S. electrostatic unit is . 

Now is clearly the value in centims. per second of 

that velocity which would be denoted by unity in the 
new system. This is a definite concrete velocity ; and its 
numerical value will always be equal to the ratio of the 
electromagnetic to the electrostatic unit of quantity, 
whatever units of length, mass, and time are employed. 

177. It will be observed that the ratio of the two units 
of quantity is the inverse ratio of their dimensions ; and 



XL] ELECTRICITY. 145 

the same can be proved in the same way of the other 
four electrical elements. The last column of the above 
table shows that M does not enter into any of the 
ratios, and that L and T enter with equal and opposite 
indices, showing that all the ratios depend only on the 

velocity -. 

Thus, if the concrete velocity be a velocity of v 

centims. per second, the following relations will subsist 

between the C.G.S. units : 

1 electromagnetic unit of quantity = v electrostatic units. 

1 current =v 

1 ,, capacity =v 2 

v electromagnetic units of potential = 1 electrostatic unit. 

v 2 resistance = 1 ,, 

178. Weber and Kohlrausch, by an experimental 
comparison of the two units of quantity, determined the 
value of v to be 

3 '1074 x 10 10 centims. per second. 

Sir. W. Thomson, by an experimental comparison of the 
two units of potential, determined the value of v to be 

2-825 x 10 10 . 

Professor Clerk Maxwell, by an experiment in which 
an electrostatic attraction was balanced by an electro- 
dynamic repulsion, determined the value of v to be 

2-88 x 10 10 . 

Professors Ayrton and Perry, by measuring the capacity 
K 



146 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

of an air-condenser both electromagnetically and statically 
("Nature," Aug. 29, 1878, p. 470), obtained the value 

2-98 x 10 10 . 

Professor J. J. Thomson (" Phil. Trans.," 1883, June 21), 
by comparing the electrostatic and electromagnetic mea- 
sures of the capacity of a condenser, and employing Lord 
Rayleigh's latest value of the B.A. resistance coils, de- 
termined v to be 

2-963 xlO 10 . 

All these values agree closely with the velocity of light 
in vacuo, of which the best determinations are, some of 
them a little less, and some a little greater than 

3 x 10 10 . 
We shall adopt this round number as the value of v. 

179. The dimensions of the electric units are rather 
simpler when expressed in terms of length, density, and 
time. 

Putting D for density, we have M = L 3 D. Making 
this substitution for M, in the expressions above obtained 
{ 174), we have the following results : 

Electrostatic. . Electromagnetic. 

Quantity, D^T- 1 D*L 2 

Current, D*L 3 T- 2 D^T- 1 

Capacity, L L~ 1 T* 

Potential, D^T- 1 D*L 3 T-2 

Resistance, L~ X T LT- 1 

It will be noted that the exponents of L and T in these 
expressions are free from fractions. 



XT.] 



ELECTRICITY. 



147 



Specific Inductive Capacity. 

180. The specific inductive capacity of an insulating 
substance is the ratio of the capacity of a condenser in 
which this substance is the dielectric to that of a conden- 
ser in other respects equal and similar in which air is the 
dielectric. It is of zero dimensions, and its value exceeds 
unity for all solid and liquid insulators. 

According to Maxwell's electro-magnetic theory of light, 
the square root of the specific indue tive capacity is equal 
to the index of refraction for the rays of longest wave- 
length. 

Messrs. Gibson and Barclay, by experiments performed 
in Sir W. Thomson's laboratory ("Phil. Trans.," 1871, 
p. 573), determined the specific inductive capacity of solid 
paraffin to be 1-977. 

Dr. J. Hopkinson (" Phil. Trans.," 1877, p. 23) gives the 
following results of his experiments on different kinds of 
flint glass : 



Kind of 
Flint Glass. 


Density. 


Specific 
Inductive 
Capacity. 


Quotient 
by 
Density. 


Index of 
Refraction 
for D line. 


Very light, ... 
Light, 


2-87 
3-2 


6-57 
6-85 


2'29 
2-14 


1-541 
1-574 


Dense, 


3 '66 


7 '4 


2'02 


1-622 


Double extra ) 
dense, \ 


4-5 


10-1 


2-25 


1-710 



In a later series of experiments ("Phil. Trans.," 1881, 
Dec. 16), Dr. Hopkinson obtains the following mean 
determinations : 



148 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

Specific Specific 

Glass. Inductive Density. . Inductive 

Capacity. Capacity. 

Hardcrown, ............ 6'96 2'485 Paraffin, 2 '29 

Very light flint, ......... 6'61 2'87 

Light flint, .............. 6'72 3'2 

Dense flint, ............... 7'38 3'66 

Double extra-dense flint, 9 "90 4 '5 

Plate, ..................... 8-45 

181. For liquids Dr. Hopkinson ("Proc. Roy. Soc.," 
Jan. 27, 1881) gives the following values of /r^ (computed) 
and K (observed), K denoting the specific inductive 
capacity and /A W the index of refraction for very long waves 
deduced by the formula 



where 6 is a constant. 

& K 

Petroleum spirit (Field's), ........ ....... 1 '922 1 '92 

Petroleum oil (Field's), .................. 2'075 2'07 

,, (common), .................. 2'078 2'10 

Ozokerit lubricating oil (Field's), ...... 2'086 2'13 

Turpentine (commercial), .............. 2'128 2'23 

Castor oil, .................................... 2-153 4'78 

Sperm oil, ................................... 2'135 3 '02 

Olive oil, .................................... 2-131 3'16 

Neatsfoot oil, .............. '. .................. 2-125 3'07 

This list shows that the equality of // to K (which 

Maxwell's theory requires) holds nearly true for hydro- 

carbons, but not for animal and vegetable oils. 

182. Wiillner (" Sitzungsber. konigl. bayer. Akad.," 

March, 1877) finds the following values of specific inductive 

capacity : 

Paraffin, ...... 1'96 Shellac, ... 2-95 to 373 

Ebonite, ...... 2-56 Sulphur, ... 2'88 to 3'21 

Plate glass, ... 6 -10 



XL] ELECTRICITY. 149 

Boltzmann ("Carl's Repertorium," x. 92165) finds 
the following values : 

Paraffin, 2'32 Colophonium, ... 2'55 

Ebonite, 3'15 Sulphur, 3'84 

Schiller ("Pogg. Ann.," clii. 535, 1874) finds: 
Paraffin. ... 5 "83 to 2 '47 Caoutchouc, ... 2'12 to 2'34 
Ebonite^ ... 2'21 to 2'76 Do., vulcanized, 2 '69 to 2 '94 

Plate glass, 5 '83 to 6 '34 

Silow (" Pogg. Ann.," clvi. and clviii.) finds the following 
values for liquids : 

Oil of turpentine, 2-155 to 2'221 

Benzene, 2'199 

Petroleum, 2*039 to 2'07l 

Boltzmann ("Wien. Akad. Ber." (2), Ixx. 342, 1874) 
finds for sulphur in directions parallel to the three princi- 
pal axes, the values 

4-773, 3-970, 3-811. 

183. Quincke (" Sitz. Pretiss. Akad.," Berlin, 1883) has 
made the following determinations. To explain the last 
two columns it is to be observed that, according to Max- 
well's theory, the charging of a condenser produces tension 
(or diminution of pressure) in the dielectric along the lines 
of force, and repulsion (or increase of pressure) perpen- 
dicular to the lines of force, the tension and the repulsion 
being each equal to 

K(A-B) 2 

" 87TC 2 

where K denotes the specific inductive capacity of the 
dielectric, c the distance between the two parallel plates 
of the condenser, and A - B their difference of potentials. 
Quincke observed the tension and repulsion, and computed 



150 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

K from each of them separately. The results are given 
in the last two columns, and are in every case greater 
than the " observed " value of K obtained in the usual 
way by comparison of capacities. 

The temperature printed below the index of refraction 
is the temperature at which the electrical experiments 
were performed. 

Density Index of Specific Computed 

aiid refraction inductive / \ 

tempera- and tern- capacity from from 

ture. perature. observed, tension, repulsion. 



I 


1-7205 


1-3605 


3-364 


4-851 


4-672 


Ether, j 


at 14 -9 


6 -60 








J 




1-3594 


3-322 


4-623 


4-660 


" I 




8 -37 








5 vols. ether to 1 bisul- ( 


8134 


1 -4044 


2-871 


4-136 


4-392 


phide of carbon, / 


16 -4 


8 -50 








1 ether to 1 bisulphide, ... j 


9966 
16 -6 


1-4955 
10 -50 


2-458 


3-539 


3-392 


1 ether to 3 bisulphide, ... j 


1-1360 
17'4 


1-5677 
5 -30 


2-396 


3-132 


3-061 


Sulphur in bisulphide of ( 


1 -3623 


1-6797 


2-113 


2-870 


2-895 


carbon (19 '5 per cent.) j 


12-6 


8 -68 








Bisulphide of carbon from $ 


1-2760 


1-6386 


2-217 


2-669 


2-743 


Kahlbaum, j 


12 -2 


7 -50 








Bisulphide of carbon from ( 


1-2796 


1 -6342 


1-970 


2-692 


2-752 


Heidelberg, j 


10 -2 


12-98 








1 vol. bisulphide to 1 tur- ( 


1-0620 


1-5442 


1-962 


2-453 


2-540 


pentine, \ 


17'8 


10'92 








Heavy benzol from ben- ( 


8825 


1 -5035 


1-928 


2-389 


2-370 


zoic acid, \ 


15 -91 


13 -20 








Pure benzol from benzoic ( 


8822 


1-5050 


2-050 


2-325 


2-375 


acid, I 


17'64 


14 -40 








Light benzol, j 


7994 
17'20 


1-4535 

ir-eo 


1-775 


2-155 


2-172 


Rape oil, j 


9159 
16 -4 


1-4743 
16 -41 


2-443 


2-385 


3-296 


Oil of turpentine, J 


8645 


1-4695 
16-71 


1-940 


2-259 


2-356 


Rock oil 


8028 


1-4483 


1-705 


2-138 


2-149 




17'0 


16 -62 









XL] ELECTRICITY. 151 

184. Professors Ayrton and Perry have found the 
following values of the specific inductive capacities of 
gases, air being taken as the standard : 






Air, 1-0000 

Vacuum, 0'99S5 

Carbonic acid, . . 1 "0008 



Hydrogen, 0'999S 

Coal gas, T0004 

Sulphurous acid, 1 '0037 



Practical Units. 

185. The unit of resistance chiefly employed by practical 
electricians is the Ohm, which is theoretically denned as 

10 9 C.G.S. electro-magnetic units of resistance. 
The practical unit of electro-motive force is the Volt, 
which is defined as 

. 10 8 C.G.S. electro-magnetic units of potential. 
The practical unit of current is the Ampere, It is de- 
fined as 

T T o of the C.G.S. electro : magnetic unit current, 
or as the current produced by 1 volt through 1 ohm. 

The practical unit of quantity of electricity is the 
Coulomb. It is defined as 

TO of the C.G.S. electro-magnetic unit of quantity, 
or as the quantity conveyed by 1 ampere in 1 second. 

The practical unit of capacity is the Farad.* It is 
defined as 

10~ 9 of the C.G.S. electro-magnetic unit of capacity, 
or as the capacity of a condenser which holds 1 coulomb 
when charged to 1 volt. 

* As the farad is much too large for practical convenience, its 
millionth part, called the microfarad, is practically employed, 
and condensers are in use having capacities of a microfarad and 
its decimal subdivisions. The microfarad is 10~ 15 of the C.G.S. 
electromagnetic unit of capacity. 



152 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

The practical unit of work employed in connection with 
these is the Joule. It is denned as 

10 7 ergs, 

or as the work done in 1 second by a current of 1 ampere 
in flowing through a resistance of 1 ohm. 

The corresponding practical unit of rate of working is 
the Watt. It is defined as 

10 7 ergs per second, 

or as the rate at which work is done by 1 ampere flowing 
through 1 ohm. 

186. The standard resistance-coils originally issued in 
1865 as representing what is now called the ohm, were 
constructed under the direction of a Committee of the 
British Association, and their resistance was generally 
called the B. A. unit. The latest and best determinations 
by Lord Rayleigh and others have shown that it was 
about 1 or, more exactly, 1 '3 per cent. too small, the 
actual resistance of the original B. A. coils being 

987 x 10 9 C.G.S. 

187. An earlier unit in use among electricians was 
Siemens' unit, defined as the resistance at C. of a 
column of pure mercury 1 metre long and 1 sq. millimetre 
in section. The resistance of such a column is about 

943 x 10 9 C.G.S. 
The reciprocal of -943 is 1 -06. 

188. The question of what electrical units should be 
adopted received great attention at the International 
Congress of Electricians at Paris in 1881 ; and the follow- 
ing resolutions were adopted : 



XL] ELECTRICITY. 153 

Resolutions adopted by the International Congress of 
Electricians at the sitting of September 22nd, 1881. 

1. For electrical measurements, the fundamental units, 
the centimetre (for length), the gramme (for mass), and 
the second (for time), are adopted. 

2. The ohm and the volt (for practical measures of 
resistance and electromotive force or potential) are to 
keep their existing definitions, 10 9 for the ohm, and 10 s 
for the volt. 

3. The ohm is to be represented by a column of 
mercury of a square millimetre section at the temperature 
of zero centigrade. 

4. An International Commission is to be appointed to 
determine, for practical purposes, by fresh experiments, 
the length of a column of mercury of a square millimetre 
section which is to represent the ohm. 

5. The current produced by a volt through an ohm is 
to be called an ampere. 

6. The quantity of electricity given by an ampere in 
a second is to be called a coulomb. 

7. The capacity defined by the condition that a coulomb 
charges it to the potential of a volt is to be called a farad. 

189. At a subsequent International Conference at Paris 
in 1884, it was agreed to define the "legal ohm" as 
" the resistance of a column of mercury 106 centimetres 
long and 1 sq. millimetre in section, at the temperature of 
melting ice." 

The following summary of experimental results was 
laid before this Conference. The two columns of numerical 
values are inversely proportional, their common product 
being 100. One of them gives the value of Siemens' 



154 



UNITS AND PHYSICAL CONSTANTS. [CHAP. 



unit in terms of the theoretical ohm (10 9 C.G.S.), and 
the other gives the length of a column of pure mercury at 
C., 1 square millimetre in section, which has a resist- 
ance of 1 theoretical ohm. 







Siemens' 


Column of 




Year. 


Observer. 


Unit in 


Mercury 


Method. 






Ohms. 


cm. 




1864. 


British Assoc. Com., 


9539 


104-83 


Brit. Association. 


1881. 


Rayleigh & Shuster, 


9436 


105-98 


Do. 


1882. 


Rayleigh, 


9410 


106-28 


Do. 


1882. 


H.Weber, 


9421 


106-14 


Do. 


1874. 


Kohlrausch, 


9442 


105-91 


Weber( 1st method). 


1884. 


Mascart, 


9406 


106-32 


Do. 


1884. 


Wiedemann, 


9417 


106-19 


Do. 


1878. 


Rowland, 


9453 


105-79 


Kirchhoff. 


1882. 


GlazeLrook, 


9408 


106-30 


Do. 


1884. 


Mascart, 


9406 


106-32 


Do. 


1884. 


F.Weber, 


9400 


105-37 


Do. 


1884. 


Roiti, 


9443 


105-90 


Roiti. 


1873. 


Lorenz, 


9337 


107-10 


Lorenz. 


1884. 


Lorenz, 


9417 


106-19 


Do. 


1883. 


Rayleigh, 


9412 


106-24 


Do. 


1884. 


Zenz, 


9422 


106-13 


Do. 


1882. 


Dora, 


9482 


105-46 


Weber (damping). 


1883. 


Wild, 


9462 


105-68 


Do. 


1884. 


H. F. Weber, 


9500 


105-26 


Do. 


1866. 


Joule, 


9413 


106-23 


Joule. 




Mean, 


9430 


106-04 





Several of the most distinguished physicists present 
expressed their opinion that 106"2 or 106'25 centimetres 
was the most probable value of the required length ; but 
in order to obtain unanimity it was agreed to adopt the 
length 106 centimetres, as above stated. 

1DO. By way of assisting the memory, it is useful to 
remark that the numerical value of the ohm is the same 



XI.] 



ELECTRICITY. 



155 



as the numerical value of a velocity of one earth-quadrant 
per second, since the length of a quadrant of the meridian 
is 10 9 centims. This equality will subsist whatever funda- 
mental units are employed, since the dimensions of resist- 
ance are the same as the dimensions of velocity. 

No special names have as yet been assigned to any 
electrostatic units. 

Electric Spark. 

191. Sir W. Thomson has observed the length of spark 
between two parallel conducting surfaces maintained at 
known differences of potential, and has computed the 
corresponding intensities of electric force by dividing (in 
each case) the difference of potential by the distance, 
since the variation of potential per unit distance measured 
in any direction is always equal to the intensity of the 
force in that direction. His results, as given on page 258 
of " Papers on Electrostatics and Magnetism," form the 
first two columns of the following table : 



Distance 
between 
Surfaces. 


Intensity of 
force in 

Electi-ostatic 
Units. 


Difference of Potential between Surfaces. 


In Electrostatic 
Units. 


In Electromagnetic 

Units. 


0086 


267-1 


2-30 


6-90xl0 10 


0127 


257-0 


3-26 


9-78 


0127 


262-2 


3-33 


9-99 


0190 


224-2 


4-26 


1278 


0281 


200-6 


5'64 


16-92 


040S 


151-5 


6-18 


18-54 


0563 


144-1 


8-11 


24-33 


0584 


139-6 


8-15 


2445 


0688 


140-8 


9-69 


29-07 


0904 


134-9 


12-20 


3660 


1056 


132-1 


13-95 


41-85 


1325 


131-0 


17-36 


2-08 



156 



UNITS AND PHYSICAL CONSTANTS. [CHAP. 



The numbers in the third column are the products of 
those in the first and second. The numbers in the 
fourth column are the products of those in the third by 
3 x 10 10 . 

192. Dr. Warren De La Rue, and Dr. Hugo W. Miiller 
("Phil. Trans.," 1877) have measured the striking dis- 
tance between the terminals of a battery of choride of 
silver cells, the number of cells being sometimes as great 
as 11000, and the electromotive force of each being 1*03 
volt. Terminals of various forms were emplo} T ed ; and 
the results obtained with parallel planes as terminals have 
been specially revised by Dr. De La Rue for the present 
work. These revised results (which were obtained by 
graphical projection of the actual observations on a larger 
scale than that employed for the Paper in the Philosophi- 
cal Transactions) are given below, together with the data 
from which they were deduced : 



DATA. 





Striking Distance. 




In Inches. 


In Centims. 


1200 


0-012 


0-0305 


2400 


021 


0533 


3600 


033 


0838 


4800 


049 


1245 


5880 


058 


1473 


6960 


073 


1854 


8040 


088 


2236 


9540 


110 


2794 


11000 


133 


3378 



XL] 



ELECTRICITY. 



157 



DEDUCTIONS. 









Intensity of Force 


Electromotive 
Force in 
Volts. 


Striking 
Distance in 
Centirns. 


Volts per 
Centim. 


In C.G.S. units. 


Electromagnetic. 


Electro- 
static. 


1000 


0205 


48770 


4-88 x 10 12 


163 


2000 


0430 


46500 


4-65 


155 


3000 


0660 


45450 


4-55 


152 


4000 


0914 


43770 


4-38 


146 


5000 


1176 


42510 


4-25 


142 


6000 


1473 


40740 


4-07 


136 


7000 


1800 


38890 


3-89 


130 


8000 


2146 


37280 


3-73 


124 


9000 


2495 


36070 


3-6! 


120 


10000 


2863 


34920 


3-49 


116 


11000 


3245 


33900 


3-39 


113 


11330 


3378 


33460 


3-35 


112 



193. The resistance of a wire (or more generally of 
a prism or cylinder) of given material varies directly as 
its length, and inversely as its cross section. It is there- 
fore equal to 

-p length 
section 

where R is a coefficient depending only on the material. 
R is called the specific resistance of the material. Its 

reciprocal is called the specific conductivity of the 
Jti 

material. 

R is obviously the resistance between two opposite 
faces of a unit cube of the substance. Hence in the C.G.S. 
system it is the resistance between two opposite faces of 
a cubic centim. (supposed to have the form of a cube). 

The dimensions of specific resistance are resistance x 
length ; that is, in electromagnetic measure, velocity x 
length ; that is, 



158 



UNITS AND PHYSICAL CONSTANTS. [CHAF. 



RESISTANCE. 

194. The following table of specific resistances is 
altered from that given in former editions of this work 
by subtracting 1'SS per cent, from all the numbers in the 
column headed " Specific Resistance," this being the correc- 
tionrequired to reduce the resistanceof mercury from 96146, 
the value previously given, to 94340, which is the value 
resulting from the new definition of the " legalohm " : 
Specific Resistances in Electromagnetic Measure 
(at C. unless otherwise stated). 



I 


Specific 
Resistance. 


Percentage 
variation per 
degree at 
'20 C. 


Specific 
Gravity. 


Silver hard-drawn, 


1579 


377 


10*50 


CoDper, 


1611 


388 


8-95 


Gold, ,, 


2114 


365 


19-27 


Lead, pressed, 


19474 


387 


11-391 


Mercury liquid, 


94340 


072 


13-595 


Gold 2, Silver 1, hard or\ 
annealed, / 


10781 


065 


15-218 


Selenium at 100 C., crys-\ 
talline / 


5-9xl0 13 


1-00 




Water at 22 C., 


7'05xl0 10 


47 




,, with -2 percent.il S0 4 


4-39 


47 




8'3 


3-26 


653 




20 


1-41 


799 




n >? 35 ,, , 


1-24 


1-259 




41 


1-34 


1-410 




Sulphate of Zinc and Water \ 
ZnS0 4 + 23H 2 Oat23C.,/ 


1-83 






Sulph. of Copper and Water) 
CuS0 4 + 45 H 2 at 22 C., / 


1-91 






Glass at 200 C. , 


2-23xl0 16 






,, 250, 


l-36x!0 15 






300, 


l-45x!0 14 






, 400 


7-21xl0 13 






Gutta Percha at 24 C. , 


3'46xl0 23 






oc., 


6-87 x 10 24 







XI.] 



ELECTRICITY. 



159 



For the authorities on which this table is based see 
Maxwell, "Electricity and Magnetism," vol. i., last 
chapter. 

195. The following table of specific resistances of 
metals at C. is reduced from Table IX. in Jen kin's 
Cantor Lectures. It is based on Matthiessen's experi- 
ments. A deduction of 1 '88 per cent, has been made, as 
in the preceding table : 





Specific 
Resistance. 


Percentage 
of Variation 
for a degree 
at 20 C. 


Resistance in i 
Ohms of a 
Wire of 1 mm. 
diam. 
1 m. long. 


Silver, annealed, 
, , hard-drawn, 
Copper, annealed, 
,, hard-drawn, 
Gold, annealed, 


1492 
1620 
1584 
1620 
2041 


377 
388 
365 


0190 
0206 
0202 
0206 
0260 


,, hard-drawn, 
Aluminium, annealed, 
Zinc, pressed, 


2077 
2889 
5581 


365 


0264 
0368 
'0749 


Platinum, annealed, 
Iron, annealed, 


8982 
9638 




1144 
1227 


Nickel, annealed, 
Tin, pressed, 
Lead, pressed, 


12358 
13103 
19468 


365 

387 


1573 
1668 
2479 


Antimony, pressed, 
Bismuth, pressed,. 


35209 
130098 


389 
354 


4483 
1 '6565 


Mercury, liquid, 


94340 


072 


1-2012 


Alloy, 2 parts Platinum, 1 | 
part Silver, by weight, j- 
hard or annealed, J 


2419 


031 


0308 


German Silver, hard or an- ^ 
nealed, j 


20763 


044 


2644 


Alloy, 2 parts Gold, 1 Sil- ^ 
ver, by weight, hard or |- 
annealed, ... J 


10779 


065 


1372 











160 UNITS AND PHYSICAL CONSTANTS. [CHAI-. 

Resistances of Conductors of Telegraphic Cables per 
nautical mile, at 24 C., in C.G.S. units. 

Red Sea, 7'79x 10 9 

Malta- Alexandria, mean, 3 '42 ,, 

Persian Gulf, mean, 6'17 ,, 

Second Atlantic, mean, 4'19 ,, 

190. The following formulae are given by Benoit* for the 
ratio of the specific resistance at t C. to that at C. : 

Aluminium, 1 + '003S76* + '000001320*- 

Copper, 1 + -00367* + '000 000 5S7* 2 

Iron, 1 + "004516* + '000 005 828* 2 

Magnesium, 1 + -003870* + '000 000 S63* 2 

Silver, 1 + '003972* + '000 000 687 * 2 

Tin, 1 + -004028* + "000 005 826* 2 

Mercury in glass tube, ^ 

apparent resistance, not M + '0008649* + '000 001 12* 2 
corrected for expansion, J 

Adopting the formula 1 + at for the ratio of the specific 
resistance at t to that at 0, MM. Cailletet and Bouty 
("Jour, de Phys.," July, 1885) have made the following 
determinations of the coefficient of variation a at very 
low temperatures : 

Range of Coefficient of 

Temperature. Variation. 

Aluminium, +28 to- 91 '00385 

Copper, -23 to-123 '00423 

Iron, to- 92 '0049 

Magnesium, to- 88 '00390 

Mercury, - 40 to - 92 '00407 

Silver, +30 to -102 '00385 

Tin, to- 85 '00424 

The new alloy called platinoid (consisting of German 
silver with a little tungsten) has been found by Mr. J. T. 

*Benoit, "Etudes expe'rimentales sur la Resistance electrique 
sous I'lnfluence de la Temperature." Paris, 1873. 



XI.] 



ELECTRICITY. 



161 



Bottoraley ("Proc. Roy. Soc.," May 7, 1885) to have an 
average variation of resistance with temperature of only 
'022 per cent, per degree centigrade, between C. and 
100 C., being about half the variation of German silver. 
Its specific resistance ranges in different specimens from 
2-9xlO- 5 to3-7xlO- 5 C.G.S. 

Resistances of Liquids. 

197. The following tables of specific resistances of 
solutions are from the experiments of Ewing and Macgregor 
("Trans. Roy. Soc., Ed in.," xxvii. 1873) : 

Solutions at 10 C. Specific Resistance. 

Sulphate of Zinc, saturated, 3 "37 x 10 10 

,, ,, minimum, 2*83 ,, 

Sulphate of Copper, saturated, 2 '93 ,, 

Sulphate of Potash, ,, T66 ,, 

Bichromate of Potash, 2 '96 ,, 

The following table is for solutions of sulphate of cop- 
per of various strengths. The first column gives the 
ratio by weight of the crystals to the water in which they 
are dissolved : 



Strength. 


Density 
at 10 C. 


Specific 
Resistance. 


Strength. 


Density. 


Specific 
Resistance. 


1 to 40 


1-0167 


16-44xl0 10 


1 to 4-146 


1-1386 


3-5 xlO 10 


30 


1-0216 


1348 , 


4 


1-1432 


3-41 


20 


1 -0318 


9-87 , 


3-297 


1-1679 


3-17 ,, 


10 


1-0622 


590 , 


3 


1-1823 


3-06 ,, 


7 
5 


1 -0858 
1-1174 


4-73 , 
3-81 , 


2-597\ 
saturated / 


1-2051 


2-93 


The following table is for solutions of sulphate of zinc: 


Strength. 


Density. 


Specific 
Resistance. 


Strength. 


Density 
at 10. 


Specific 
Resistance. 


1 to 40 


1-0140 


18-29xl0 10 


1 to 3 


1-1582 


337x10 


20 


1-0-J78 


11-11 ,, 


2 


1-2186 


803 ,, 


10 


1-0540 


6-38 


1-5 


1 "270 


285 


-7 


1 -0760 


5-08 


1 


1 3530 


3 10 ,, 


5 


1-1019 


4-21 


752 } 
saturated/ 


1 -4220 


3-37 






L 







162 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

The following table for dilute sulphuric acid is from 
Becker's experiments, as quoted by Jamin and Bouty, 
torn. iv. p. Ill : 

Specific Resistance 



Density. 


/^At 0. 


At 8. 


At 16. 


At 24. ^ 


MO 


1-37 xlO 10 


1-04 xlO 10 


845 x 10 10 


737 


xlO 10 


1-20 


1-33 ,, 


926 


666 


486 


5) 


1-25 


1-31 


896 


624 


434 


55 


1-30 


1-36 ,, 


94 


662 


472 


,, 


1-40 


1-69 ,, 


1-30 


1-05 


896 


?) 


1-50 


2-74 


2-13 


1-72 


1-52 


J? 


1-60 


4-82 ,, 


3-02 


275 


2-21 


,, 


1-70 


9-41 


6-25 


4-23 ,, 


3-07 


,, 



Resistance of Carbons. 

198. The specific resistance of Carre's electric-light 
carbons at 20 C. is stated to be 

3-927 x 10 6 C.G.S., 

whence it follows that the resistance of a cylinder 1 metre 

long and 1 centimetre in diameter is just half an ohm. 

The specific resistance of Gaudin's carbons is about 8 '5 x 10 6 
,, ,, retort carbon 6*7 x 10 7 

graphite from 2'4 x 10 6 to4'2 x 10 r 

The resistance of carbon diminishes as the temperature 

increases, the diminution from to 100 C. being for 

JL \) 

Carre's and for Gaudin's. The resistance of an incan- 
descent lamp when heated as in actual use is about half 
its resistance cold. 






XL] ELECTRICITY. 163 

Resistance of the Electric Arc. 

199. The difference of potentials between the two 
carbons of an arc lamp has been found by Ayrton and 
Perry ("Phil. Mag.," May, 1883) to be practically in- 
dependent of the strength of the current, when the dis- 
tance between them is kept constant. It was scarcely 
altered by tripling the strength of the current. The 
apparent resistance of the arc (including the effect of 
reverse electromotive force) is therefore inversely as the 
current. The difference of potentials was about 30 volts 
when the current was from 6 to 12 amperes. 

200. The following approximate determinations of the 
resistance of water and ice at different temperatures are 
contained in a paper by Professors Ayrton and Perry, 
dated March, 1877 (" Proc. Phys. Soc., London," vol. ii. 
p. 178):- 

Temp. Specific 

Cent. Resistance. 

-12-4 2'240xl0 18 

- 6-2 1-023 

- 5-02 9-4S6xl0 17 

- 3'5 6-428 

- 3-0 5-693 , 



about 



2-46 


4'844 ,, 


- 1-5 


3-876 ,, 


- 0-2 


2-840 , 


+ 0-75 


1-188 ,, 


+ 2-2 


2-48 x 10 16 


+ 4-0 


9-1 x 10 15 


+ 7-75 


5-4 x 10 14 



+ 11-02 , 3-4 



The values in the original are given in megohms, and 
we have assumed the megohm = 10 15 C.G-.S. units. 



164 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

According to F. Kohlrausch (" Wied. Ann.," xxiv. 
p. 48, 1885) the resistance at 18 C. of water purified by 
distillation in vacuo is 4 x 10 10 times that of mercury. 
This makes its specific resistance 

3-76 x 10 15 . 

201. The specific resistance of glass of various kinds at 
various temperatures has been determined by Mr. Thomas 
Gray (" Proc. Roy. Soc.," Jan. 12, 1882). The following 
are specimens of the results : 

Bohemian Glass Tubing, density 2*43. 

At 60 6-05 xlO- 2 At 160 2'4 x 10 19 

100 2 x 10- 1 174 8-7 x 10 18 

130 2 x 10 20 

Thomson's Electrometer Jar (flint glass), density 3 '172. 

At 100 2-06 x 10 23 At 160 2'45 x 10- 1 

120 4-6Sxl0 2 - 180 5-6 x 10 20 

140 1-06 200 1-2 

The following are all at 60 C. : 

Bohemian Beaker, 4 '25 x 10 22 density 2 '427 

7-15 ,, 2-587 

Florence Flask, 4 '69 x 10 20 2'523 

Test Tube, 1'44 ,, 2435 

3-50 2-44 

Flint Glass Tube, 3*89 x 10 22 2753 
Thomson's Electro- ^ 

meter Jar (flint [ 1 '02 x 10 24 3'172 
glass), J 

202. The following appoximate values of the specific 
resistance of insulators after several minutes' electrifi- 
cation are given in a paper by Professors Ayr ton and 
Perry (-'Proc. Royal Society," March 21, 1878), "On 
the Viscosity of Dielectrics " : 



XI.J 



ELECTRICITY. 



165 



Specific Temperature. 
Resistance. Centigrade. 






Authority. 

Ayrton and Perry. 
(Standard adopted by 
^ Latimer Clark. 

Ayrton and Perry. 

Recent cable tests. 

Ayrton and Perry. 



Mica, 8'4xl0 22 20 

Gutta-Percha, 4 '5 x 10 23 24 

Shellac, 9'OxlO 24 28 

Hooper's Material, 1 '5 x 10 25 24 

Ebonite, 2'SxlO 25 46 

Paraffin, 3'4xl0 25 46 

pi f Not yet measured with accuracy, but greater 

' \ than any of the above. 

Air, Practically infinite. 

203. Particulars of Board of Trade Standard Gauge 
of Wires (Imperial Gauge) Nos. 4 to 20. 



No. 


Diameter. 


Sectional 
area. 

Sq. inches. 


Resistance in ohms of 1 metre 
length pure copper at C. 


Milli- 
metres. 


Thou- 
sandths 
of inch. 


Annealed. 


Hard-drawn. 


4 


5-89 


232 


04227 


0005929 


0006065 


5 


5-38 


212 


03530 7107 


7269 


6 


4-88 


192 


02895 


8638 


8835 


7 


4-47 


176 


02433 


001029 


001053 


8 


4-06 


160 


0201 1 


1248 


1276 


9 


3-66 


144 


01629 


1536 


1571 


10 


3-25 


128 


012^7 


1948 


1992 


11 


2-95 


116 


01057 


2364 


2418 


12 


2-64 


104 


008494 


2951 


3019 


13 


234 


92 


006647 


3757 


3842 


14 


203 


80 


005026 


4992 


5106 


15 


1-83 


72 


004070 


6142 


6283 


16 


163 


64 


003216 


7742 


7919 


17 


1-42 


56 


002463 


01020 '01043 


18 


1-22 


48 


001809 


01382 


01414 


19 


1-016 


40 


001256 


01993 


02038 


20 


0914 


36 


000917 


024152 


02518 








! 



166 



UNITS AND PHYSICAL CONSTANTS. [CHAP. 



The heat generated per second in 1 metre length of 



pure copper wire at C. 



is -0048(0) 



=-) gin. deg., and 



at 40 C. is -0055J J gm. deg., 0. denoting the current 
in amperes, and D the diameter in millimetres. 

204. Resistance of 1 metre length of Wires of Imperial 
Gauge at C. (For copper see preceding table.) 



No. 


Iron, 
annealed. 


German Silver, 
either 
annealed or 
hard-drawn. 


Platinum, 
annealed. 


Silver, annealed. 


4 


003606 


007768 -003361 


0005583 


5 


4322 


9311 4028 


6692 


6 


5253 


01132 4896 


8184 


7 6261 


01349 


5836 


9694 


8 


7590 


01635 7074 


001175 


9 


9339 


02012 8705 


1446 


10 


01184 


02552 -01104 


1834 


11 


01438 


03096 -01340 


2226 


12 


01795 


03867 '01673 


2779 


13 


02285 


04922 -02129 


3538 


14 


03036 


06540 -02829 


4700 


15 


03736 


08047 -03482 


5784 


16 


04708 


1014 


04388 


7290 


17 '06204 


1336 -05782 


9606 


18 -08405 


1811 


07834 


01301 


19 


1212 


2611 -1130 


01876 


20 -1498 


3226 


1396 


02319 



Electromotive Force. 

205. The electromotive force of a Daniell's cell was 
found by Sir W. Thomson (p. 245 of "Papers on Electricity 
and Magnetism ") to be 

00374 electrostatic unit, 



XT.] ELECTRICITY. 167 

from observation of the attraction between two parallel 
discs connected with the opposite poles of a Daniell's 
battery. As 1 electrostatic unit is 3 x 10 10 electromag- 
netic units, this is -00374 x 3 x 10 10 = M22 x 10 s electro- 
magnetic units, or 1-122 volt. 

According to Latimer Clark's experimental determina- 
tions communicated to the Society of Telegraph Engineers 
in January, 1873, the electromotive force of a Daniell's 
cell with pure metals and saturated solutions, at 64 F., 
is 1*105 volt, and the electromotive force of a Grove's 
cell 1-97 volt. These must be diminished by 1 per cent, 
because they were deduced from the assumption that the 
B. A. unit of resistance was correct. They will thus be 
reduced to 1*094 and 1*95 volts. 

According to the determination of F. Kohlrausch 
("Pogg. Ann.," vol. cxli. [1870], and Erganz., vol. vi. 
[1874], p. 35) the electromotive force of a Daniell's cell 
is 1-138 x 10 8 , and that of a Grove's cell 1-942 x 10 s . 
These must be diminished by 3 per cent., because they 
were deduced from the value '9717 x 10 9 for Siemens' 
unit which is 3 per cent, too great. They will thus be 
reduced to 1-104 and 1*884 volts. 

H. S. Carhart ("Amer. Jour. Sci. Art.," Nov. 1884) 
has found the following different values for the electro- 
motive force of a Daniell's cell according to the strength 
of the zinc sulphate solution : 

Percent. Electromotive Electromotive 

^^an. force in volts. Percent. force 



1 1-125 10 1-118 

3 1-133 15 1-115 

5 1-142 20 1-111 

74 1-120 25 1-11 1 



168 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

He finds by the same method the electromotive force of 
Latimer Clark's standard cell to be 1*434 volt. 

LordRayleigh ("Phil. Trans.," June 1884, p. 452) has 
determined the electromotive force of a Clark cell at 
15 C. to be 

1-435 volt. 

The value formerly assigned to it was 1'457 volt, and 
was based on the assumption that the B. A. unit of resist- 
ance was correct. 

In a supplementary paper (Jan. 21, 1886) he gives the 
general result for any temperature t, 

1-435(1- 0-00077(^-15)}, 

together with full particulars as to the precautions neces- 
sary for securing constancy. 

206. Professors Ayrton and Perry have made deter- 
minations of the electromotive forces called out by the 
contacts, two and two, of a great number of substances 
measured inductively. The method of experimenting is 
described in the Proceedings of the Royal Society for 
March 21, 1878. The following abstract of their latest 
results was specially prepared for this work by Professor 
Ayrton in January, 1879 : 



.] ELECTRICITY. 


ed directly by experiment, those with an asterisk by calculation, using the 
t of metals, all at the same temperature, there is no electromotive force, 
ame of a substance are the differences of potential, in volts, between that ^ 
il row as the number, the two substances being in contact. Thus ' lead is 
ict being 0'542 volts. 
3fore only commercially pure. 


(jnoq^ ) j 'Sui^uauiuadxa 
jo ounj aq:} !ye aan^'Bjaduia l j l aSfeaaAy 


sstug; 


1 i 


t- (N 05 OJ 

00 t t <M 

(N CO CO 00 

1 1 1 


-ureSi'Buiv 


* * * 


* 
O CO M< IM 

2 9 ? o S 


ou !Z 


^ -L, o 

C5 IO O i 1 


1 s o s i 


-,* 


CO CO 05 
t*** "T^ CO ^^ 

1 


O i CO C<J 

1 1 





CO QO O5 i i 
"-H C^ CO !> 
1 i 


'i i 7 "i 




1 1 1 o 


1 I 


, OJI 


1C ^O ^H 

GO ^ ^^ 
Tt< TH O T* 


O5 CO O T* Tf< 

^H O "* 

1 1 1 


The numbers without an asterisk were obtain 
well-known assumption that in a compound circu 
The numbers in a vertical column below the n 
substance and the substance in the same hoiizont 
positive to copper, the electromotive force of conts 
The metals were those of commerce, and ther 




O CO C<l 
CO O ^ ip 

1 1 


QO CO CO ^ l^* 
C< ^ !> CO O 
I 1 1 1 





O IO OO 
O CO ^ OO 

1 1 1 


* * * * 

1 ' 7 7 ' 




f i a- i 

s | | j 


1 !l| i 

^ C c gN g 
PH H N ^ PQ 



CONTACT DIFFERENCES OF POTENTIAL IN VOLTS. 





j 


1 






a 




g 

6 


1 


H 


"i 
3 


1 


1 






Mercury 


"092 


'308 


"502 




"156 








f 


01 to -17 


269 






285 








DistiUed water, ...J 


depend- 


to 


148 


171 


to 


177 






I 


arbon. 


100 






345 








Alum, saturated at ) 
16'5 C., I 


.. 


-127 


-653 


-139 


246 


-225 






Copper sulphate solu- \ 
tion, specific gravity, > 
1-087 at 16-6C., .... j 




103 

















Copper sulphate, satu- ) 
rated at 15 C., .... f 




070 


.. 










OQ 

d 


Sea salt, specific ) 
gravity, 1'18 at V 




-475 


-605 


-267 


-856 


-334 






20'5C., ) 
















O 


Sal-ammoniac, satu- ) 
rated at 15-5 C., .. f 




-396 


-652 


-189 


057 


-364 




CQ 


Zinc sulphate solu- ") 


















tion, specific grav- > 
ity, 1-125 at 16 -9 C., j 






























Zinc sulphate, satu- ) 


















rated at 15*3 C., . . I 

1 Distilled water mixed ) 
with 3 zinc sulphate, > 














saturated solution, ) 














o 


r ~j 


20 Distilled water, ) 

















*|1 


1 strong sulphuric > 
acid, ) 


























Dd 

SQ 




10 Distilled water, \ 
1 strong sulphuric > 
acid, ) 


about 
-035 
























-w / 


5 Distilled water, ) 














^ 


^""S) K 
















H 




acid j 














P 



4! 


1 Distilled water, ) 
5 strong sulphuric > 


3 to 

01 




.. 


-120 


.. 


-25 
















1 -fiOO 




CONCEN- f 
TRATED.1 


Sulphuric acid, -< 
Nitric acid 


depend 
ing on 
carbon 


1-113 




720 to 
1-252 


to 
1-300 

672 


















Mercurous sulphate ) 


















paste, I 
Distilled water, with ) 


















acid, j 

















The average temperature at the time of experimenting was about 16 C. 
All the liquids and salts employed were chemically pure ; the solids, how- 
ever, were only commercially pure. 



Solids with Liquids and Liquid* with Liquids in Air. 



1 

N 


Amalgamated Zinc. 





Mercury. 


Distilled Water. 


AlumSolution .satu- 
rated at 16 -5 C. 


ft 

OT G^> 

.2 * 
> 


Zinc Sulphate Solu- 
tion .Specific Grav- 
ity 1-125 at 16 -9 C. 


Zinc Sulphate Solu- 
tion, saturated at 
15-3 C. 


1 Distilled Water, 
3 Zinc Sulphate. 


Strong Nitric Acid. 


-105 






















to 


100 


231 








-043 




164 






+156 






















-536 




-014 










'090 
















-043 








095 


102 




-565 




-435 


















-637 




-348 


















-430 


-2S4 






-200 




-095 










-'444 












- '102 










-344 
























-358 























-429 

























016 




















848 







1-298 


1-456 


1-269 




1-609 












475 
















-241 












- 









078 



Example of the above table : Lead is positive to distilled water, and 
the contact difference of potentials is 0'171 volt. 



172 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

The authors point out that in all these experiments the 
unknown electromotive forces of certain air contacts are 
included. 

From these tables we find we can build up the electro- 
motive forces of some well-known cells. Eor example, in 
a Daniell's cell there are four contact differences of potential 
to consider, and in a Grove's cell five, viz. : 

DanieWs Cell. 

Volts. 

Copper and saturated copper sulphate, +0 '070 

Saturated copper sulphate and saturated zinc sulphate. - 0"095 

Saturated zinc sulphate and zinc, +0'430 

Zinc and copper, +0 '750 

1-155 
Grove's Cell 

Copper and platinum, + 0'238 

Platinum and strong nitric acid, + 0-672 

Strong nitric acid and very weak sulphuric acid, + 0*078 

Very weak sulphuric acid and zinc, +0 '241 

Zinc and copper, +0 '750 

1-979 
Thermoelectricity. 

207. The electromotive force of a thermoelectric circuit 
is called Thermoelectric force. It is proportional ccet. par. to 
the number of couples. The thermoelectric force of a single 
couple is in the majority of cases equal to the product of 
two factors, one being the difference of temperature of 
the two junctions, and the other the difference of the thermo- 
electric heights of the two metals at a temperature midway 
between those of the junctions. The current through the 
hot junction is from the lower to the higher metal when 
their heights are measured at the mean temperature. 



XI.] 



ELECTRICITY. 



173 



Our convention as to sign (that is, as to up and down 
in speaking of thermoelectric height) is the same as that 
adopted by Prof. Tait, and is opposite to that adopted in 
the first edition of this work. We have adopted it 
because it leads to the rule (for the Peltier and Thomson 
effects) that a current running down generates heat, and 
a current running up consumes heat. 

The following table of thermoelectric heights relative to 
lead can be employed when the mean temperature of the 
two junctions does not differ much from 19 or 20 C. 
It is taken from Jenkin's "Electricity and Magnetism," 
p. 176, where it is described as being compiled from 
Matthiessen's experiments. We have reversed the signs 
to suit the above convention, and have multiplied by 100 
to reduce from microvolts to C.G.S. units. 



Thermoelectric Heights at about 20 C. 



Bismuth, pressed com-) _ cfiQQ 
mercial wire, / 
Bismuth, pure pressed ) ggrjo 
wire, . _/" 


Antimony, pressed wire + 280 
Silver, pure hard, + 300 
Zinc, puie pressed, + 370 
Copper, ffalvano-plas- ) 


Bismuth, crystal, axial, - 6500 
, , equatorial - 4500 
Cobalt, -2200 


tically precipitated, / + 
Antimony, pressed) fift , 
commercial wire, ) 


German Silver - 1175 


Arsenic . + 1356 


Quicksilver, - 41 '8 


Iron, pianoforte wire, + 1750 


Lead, 


Antimony, axial, + 2260 


Tin, + 10 


,, equatorial, + ^640 


Copper of Commerce, ... + 10 
Platinum + 90 


Phosphorus, red, + 2970 
Tellurium, . . +50200 


Gold,.. ...+ 120 


Selenium, . . . . . + 80700 



208. The following table is based upon Professor Tait's 
thermoelectric diagram (" Trans. Roy. Soc., Edin.," vol. 
xxvii. 1873) joined with the assumption that a Grove's 
cell has electromotive force 1'97 x 10 s : 



174 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

Thermoelectric Heights at 
t" C. in C.G.S. units. 

Iron, +1734- 4'87 t 

Steel, + 1139- 3'28* 

Alloy, believed to be Platinum Iridium, + 839 at all temperatures. 

Alloy, Platinum 95 ; Iridium 5, + 622 - '55 t 

,, 90; ,, 10, + 596- T34 

85; ,, 15, + 709- '63 

,, ,, 85; ,, 15, + 577 at all temperatures. 

Soft Platinum, - 61- 1'lOt 

Alloy, platinum and nickel, + 544 - 1 *10 1 

Hard Platinum, + 260- '75 t 

Magnesium, + 244 - '95 t 

German Silver, -1207- 5'l2t 

Cadmium, + 266+ 4'29 

Zinc, + 234+ 2'40 

Silver, + 214+ l'50t 

Gold, + 283+ T02 

Copper, + 136+ "95t 

Lead, 

Tin, - 43+ -55 

Aluminium, - 77+ "39 t 

Palladium, - 625- 3'59 

Nickel to 175 C., -2204- 5'l2t 

,, 250 to 310 C., -8449 + 24-U 

,, from 340 C., - 307- 5'12 

The lower limit of temperature for the table is - 18 C. 
for all the metals in the list. The upper limit is 416 C., 
with the following exceptions : Cadmium, 258 C.; Zinc, 
373 C. ; German Silver, 175 C. 

Ex. 1. Required the electromotive force of a copper-iron 
couple, the temperatures of the junctions being C. 
and 100 C. 

We have, for iron, +1734-4 -87; 

copper, + 136+ -95*; 

iron above copper, 1598 - 5 *82 






XT. 1 ELECTRICITY. 175 

The electromotive force per degree is 
1598-5-82x50 = 1307, 
and the electromotive force of the couple is 

1307(100-0) = 130,700, 
tending from copper to iron through the hot junction. 

By the neutral point of two metals is meant the tem- 
perature at which their thermoelectric heights are equal. 

Ex. 2. To find the neutral point of copper and iron we 
have 

1598-5-82^ = 0, = 275; 

that is, the neutral point is 275 C. When the mean of 
the temperatures of the junctions is below this point, the 
current through the warmer junction is from copper to 
iron. The current ceases as the mean temperature attains 
the neutral point, and is reversed in passing it. 

Ex. 3. F. Kohlrausch (" Pogg. Ann. Erganz.," vol. vi. 
p. 35, 1874) states that, according to his determination, the 
electromotive force of a couple of iron and German silver 
is 24 x 10 5 millimetre-milligramme-second units for 1 of 
difference of temperatures of the junctions, at moderate 
temperatures. Compare this result with the above Table 
at mean temperature 100. 

The dimensions of electromotive force are M^L^T" 2 ; 

hence the C.G.S. value of Kohlrausch's unit islO'MO"^ 
= 10~ 3 , giving 2400 as the electromotive force per degree 
of difference. 

From the above table we have 

Iron above German silver, 2941 + -252, 
which, for = 100, gives 2966 as the electromotive force 
per degree of difference. 



176 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

Peltier and Thomson Effects. 

209. When a current is sent through a circuit com- 
posed of different metals, it produces in general three 
distinct thermal effects. 

1. A generation of heat to the amount per second of 
C 2 R ergs, C denoting the current, and R the resistance. 

2. A generation of heat or cold at the junctions. This 
is called the Peltier effect, and its amount per second in 
ergs at any one junction can be computed by multiplying 
the difference of thermoelectric heights at this junction 
by + 273 and by the current, t denoting the centigrade 
temperature of the junction. If the current flows down 
(that is from greater to less thermoelectric height) the 
effect is a warming ; if it flows up, the effect is a cooling. 

Ex. 4. Let a unit current (or a current of 10 amperes) 
flow through a junction of copper and iron at 100 C. 
The thermoelectric heights at 100 C. are 

Iron, 1247 

Copper, 231 

Iron above copper, 1016 

Multiplying 1016 by 373, we have about 379,000 ergs, or 
- of a gramme-degree, as the Peltier effect per second. 

Heat of this amount will be generated if the current is 
from iron to copper, and will be destroyed if the current 
is from copper to iron. 

3. A generation of heat or cold in portions of the cir- 
cuit consisting of a single metal in which the temperature 
varies from point to point. This is called the Thomson 
effect. Its amount per second, for any such portion of 



xi.] ELECTRICITY. 177 

the circuit, is the difference of the thermoelectric heights 
of the two ends of the portion, multiplied by 273 + , 
where t denotes the half-sum of the centigrade tempera- 
tures of the ends, and by the strength of the current. 

The Thomson effect, like the Peltier effect, is reversed 
by reversing the current, and follows the same rule that 
heat is generated when the current is from greater to less 
thermoelectric height. 

Experiment shows that the Thomson effect is insensible 
in the case of lead; hence the thermoelectric height of 
lead must be sensibly the same at all temperatures. It is 
for this reason that lead is adopted, by common consent, 
as the zero from which thermoelectric heights are to be 
reckoned. 

Ex. 5. In an iron wire with ends at C. and 100 C., 
the cold end is the higher (thermoelectrically) by 
4-87 x 100 that is, by 487. Multiplying this differ- 
ence by 273 + 1(0 + 100) or 323, we have 157300 as the 
Thomson effect per second for unit current. This amount 
of heat (in ergs) is generated in the iron when the current 
through it is from the cold to the hot end, and is destroyed 
when the current is from hot to cold. 

Ex. 6. In a copper wire with ends at C. and 100 C., 
the hot end is the higher by '95 x 100 or 95. Multiply- 
ing this by 323, we have 30700 (ergs) as the Thomson 
effect per second per unit current. This amount of heat 
is generated in the copper when the current through it is 
from hot to cold, and destroyed when the current is from 
cold to hot. 

The effect of a current from hot to cold is opposite in 
these two metals, because the coefficients of t in the 

M 



178 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

expressions for their thermoelectric heights (p. 174) have 
opposite signs. 

Relation between Thermoelectric Force and the Peltier and 
Thomson effects. 

210. The algebraic sum of the Peltier and Thomson 
effects (expressed in ergs) due to unit current for one second 
in a closed metallic circuit, is equal to the thermoelectric 
force of the circuit; and the direction of this thermoelectric 
force is the direction of a current round the circuit which 
would give an excess of destruction over generation of 
heat (so far as these two effects are concerned). 

Ex. 7. In a copper-iron couple with junctions at 
0. and 100 C., suppose a unit current to circulate in 
such a direction as to pass from copper to iron through 
the hot junction, and from iron to copper through the cold 
junction. 

The Peltier effect at the hot junction is a destruction 
of heat to the amount 1016 x 373 = 379,000 ergs. 

The Peltier effect at the cold junction is a generation of 
heat to the amount 1598 x 273 = 436,300 ergs. 

The Thomson effect in the iron is a destruction of heat 
to the amount 487 x 323 = 157,300 ergs. 

The Thomson effect in the copper is a destruction of 
heat to the amount 95 x 323 = 30,700 ergs. 

The total amount of destruction is 567,000, and of 
generation 436,300, giving upon the whole a destruction 
of 130,700 ergs. The electromotive force of the couple is 
therefore 130,700, and tends in the direction of the 
current here supposed. This agrees with the calculation 
in Example 1. 






XL] ELECTRICITY. 179 

Electrochemical Equivalents. 

211. The quantity of a given metal deposited in an 
electrolytic cell or dissolved in a battery cell (when there 
is no " local action ") depends on the quantity of electricity 
that passes, irrespective of the time occupied. Hence we 
can speak definitely of the quantity of the metal that is 
"equivalent to" a given quantity of electricity. By the 
electrochemical equivalent of a metal is meant the quantity 
of it that is equivalent to the unit quantity of electricity. 
In the C. G. S. system it is the number of grammes of 
the metal that are equivalent to the C.G.S. electromagnetic 
unit of electricity. 

Special attention has been paid to the electrochemical 
equivalent of silver, as this metal affords special facilities 
for accurate measurement of the deposit. The latest 
experiments of Lord Rayleigh and Kohlrausch agree in 
giving 

01118 

as the C.G.S. electrochemical equivalent of silver.* 

The number of grammes of silver deposited by 1 
ampere in one hour is 

01118 X T \J x 3600 = 4-025. 

212. The electrochemical equivalents of the most im- 
portant of the elements are given in the following table. 
They are calculated from the chemical equivalents in the 
preceding column by simple proportion, taking as basis 
the above-named value for silver. Their reciprocals are 
the quantities of electricity required for depositing one 

* Rayleigh's determination is '0111794; Kohlrausch 's, '011183; 
Mascart's, '011156. See "Phil. Trans.," 1884, pp. 439, 458. 



180 



UNITS AND PHYSICAL CONSTANTS. [CHAP. 



gramme. The quantity of electricity required for deposit- 
ing the number of grammes stated in the column " chemical 
equivalents" is the same for all the elements, namely, 
9634 C.G.S. units. 



Elements. 


Atomic 
Weight. 


Valency. 


Chemi- 
cal 
Equiva- 
lents. 


Electro- 
chemical 
equivalents 
orgrammes 
per unit of 
electricity. 


Recipro- 
cal or 
Electri- 
city per 
gramme. 


Electro-positive 
Hydrogen, 
Potassium, 


1 

39-03 
23-00 
196-2 

1077 
63-18 

199-8 
117-4 

55-88 
> 5 
58-6 
64-88 
206-4 
27-04 

15-96 
35-37 
126-54 
79-76 
14-01 


1 
1 
1 

3 
1 
2 
1 
2 
1 
4 
2 
3 
2 
2 
2 
2 
3 

2 
1 

1 
1 
3 


1 

39-03 
23-00 
65-4 
107-7 
31-59 
63-18 
99-9 
199-8 
29-35 
58-7 
18-63 
27-94 
29-3 
32-44 
103-2 
9-01 

7-98 
35-37 
126-54 
79-76 
4-67 


0001038 
004051 
002387 
006789 
01118 
003279 
006558 
01037 
02074 
003046 
006093 
001934 
002900 
003042 
003367 
01071 
000935 

0008283 
003671 
013134 
008279 
0004847 


9634 
246-9 
418-9 
147-3 
89-45 
305-0 
152-5 
96-43 
48-22 
328-3 
164-1 
517-1 
344-8 
328-7 
297-0 
93-37 
1070 

1207 
272-4 
76-14 
120'8 
2063 


Sodium, 


Gold, 


Silver, 


Copper (cupric) , 
,, (cuprous), 
Mercury (mercuric), . . . 
,, (mercurous), 
Tin (stannic), 


,, (stannous), 


Iron (ferric), . 


,, (ferrous), 
Nickel, 


Zinc, 


Lead, 


Aluminium, 


Electro-negative 
Oxygen, 


Chlorine, 
Iodine, 
Bromine, 


Nitrogen, 





To find the equivalent of 1 coulomb, divide the above 
electrochemical equivalents by 10. 

To find the number of grammes deposited per hour by 
1 ampere, multiply the above electrochemical equivalents 
by 360. 






XL] ELECTRICITY. 181 

213. Let the " chemical equivalents " in the above table 
be taken as so many grammes; then, if we denote by H 
the amount of heat due to the whole chemical action 
which takes place in a battery cell during the consumption 
of one equivalent of zinc, the chemical energy which runs 
down, namely JH ergs, must be equal (if there is no 
wasteful local action) to the energy of the current pro- 
duced. But this is the product of the quantity of 
electricity 9634 by the electromotive force of the cell. 

TTT 

The electromotive force is therefore equal to a( ^-- 



In the tables of heats of combination which are in use 
among chemists, the equivalent of hydrogen is taken as 2 
grammes, and that of zinc as 64*88 or 65 grammes. The 
equivalent quantity of electricity will accordingly be 
9634 x 2, and the formula to be used for calculating the 

TTT 

electromotive force of a cell will be -. 

19Joo 

In applying this calculation to Daniell's and Grove's 
cells, we shall employ the following heats of combination, 
which are given on page 614 of Watts' "Dictionary of 
Chemistry," vol. vii., and are based on Julius Thomsen's 
observations : 

Zn, O, SO 3 , Aq., ............... 108,462 

Cu, O, SO 3 , Aq., ............... 54,225 

N 2 2 , O 3 , Aq., ............... 72,940 

N 2 2 , O, Aq., .................. 36,340 

In Daniell's cell, zinc is dissolved and copper is set 
free, we have, accordingly, 

H = 108,462 - 54,225 = 54,237. 
In Grove's cell, zinc is dissolved and nitric acid is 



J 82 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

changed into nitrous acid. The thermal value of this 
latter change can be computed from the third and fourth 
data in the above list, as follows : 

72,940 is the thermal value of the action in which, by the 
oxidation of one equivalent of N 2 2 and combination with 
water, two equivalents of NHO 3 (nitric acid) are produced. 

36,340 is the thermal value of the action in which, by 
the oxidation of one equivalent of N 2 2 and combination 
with water, two equivalents of NHO 2 (nitrous acid) are 
produced. The difference 36,600 is accordingly the ther- 
mal value of the conversion of two equivalents of nitrous 
into nitric acid, and 18,300 is the value for the conversion 
of one equivalent. In the present case the reverse changes 
take place. We have, therefore, 

H = 108,462 - 18,300 = 90,162. 

Taking J as 4'2 x 10 7 , the value of ^JL wil1 be 

1-182 x 10 8 for Daniell's cell. 
1-985 x 10 s Grove's 

These are greater by from 2 to 8 per cent, than the direct 
determinations given in 205. 

214. Examples in Electricity. 

1. Two conducting spheres, each of 1 centim. radius, are 
placed at a distance of r centims. from centre to centre, 
r being a large number; and each of them is charged 
with an electrostatic unit of positive electricity. "With 
what force will they repel each other 1 

Since r is large, the charge may be assumed to be uni- 
formly distributed over their surfaces, and the force will 
be the same as if the charge of each were collected at its 

centre. The force will therefore be - of a dyne. 

r 2 



XL] ELECTRICITY. 183 

2. Two conducting spheres, each of 1 centim. radius, 
placed as in the preceding question, are connected one 
with each pole of a Daniell's battery (the middle of the 
battery being to earth) by means of two very fine wires 
whose capacity may be neglected, so that the capacity of 
each sphere when thus connected is sensibly equal to 
unity. Of how many cells must' the battery consist that 

the spheres may attract each other with a force of 2 of a 

dyne, r being the distance between their centres in cen- 
tims. 1 

One sphere must be charged to potential 1 and the other 
to potential - 1. The number of cells required is 



3. How many Daniell's cells would be required to pro- 
duce a spark between two parallel conducting surfaces at 
a distance of '019 of a centim., and how many at a distance 
of -0086 of a centim. 1 (See 178, 184.) 

4-26 11QO 2-30 
AnS ' <X)S7-i- 1139; .06374 = 

4. Compare the capacity denoted by 1 farad with the 
capacity of the earth. 

The capacity of the earth in static measure is equal to 
its radius, namely 6 '37 x 10 s . Dividing by v' 2 to reduce 
to magnetic measure, we have '71 x 10~ 12 , which is 1 
farad multiplied by '71 x 10~ 3 , or is '00071 of a farad. 
A farad is therefore 1400 times the capacity of the earth. 

5. Calculate the resistance of a cell consisting of a 
plate of zinc, A square centims. in area, and a plate of 
copper of the same dimensions, separated by an acid 



184 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

solution of specific resistance 10 9 , the distance between 
the plates being 1 centim. 

Ans. - , or of an ohm. 
A A 

6. Find the heat developed in 10 minutes by the 
passage of a current from 10 Daniell's cells in series 
through a wire of resistance 10 10 (that is, 10 ohms), 
assuming the electromotive force of each cell to be 
1*1 x 10 8 , and the resistance of each cell to be 10 9 . 
Here we have 

-Total. electromotive force = 1*1 x I 9 . 
Resistance in battery = 10 10 . 
Resistance in wire = 1 10 . 

Current = ^ * : * = '55 x 10' 1 = -055. 



Heat developed in ) = (-Q55 2 ) x IQiQ _ 7 . 2()94 
wire per second j 4'2 x 10 7 

Hence the heat developed in 10 minutes is 4321 '4 
gramme-degrees. 

7. Find the electromotive force between the wheels on 
opposite sides of a railway carriage travelling at the rate 
of 30 miles an hour on a line of the ordinary gauge 
[4 feet 8J inches] due to cutting the lines of force of 
terrestrial magnetism, the vertical intensity being -438. 

The electromotive force will be the product of the 
velocity of travelling, the distance between the rails, and 
the vertical intensity, that is, 

(44-7 x 30) (2-54 x 56-5) (-438) = 84,300 
electromagnetic units. 

This is about r~ of 



XL] ELECTRICITY. 185 

8. Find the electromotive force at the instant of passing 
the magnetic meridian, in a circular coil consisting of 300 
turns of wire, revolving at the rate of 10 revolutions per 
second about a vertical diameter \ the diameter of the 
coil being 30 centims., and the horizontal intensity of 
terrestrial magnetism being '1794, no other magnetic 
influence being supposed present. 

Self-induction can be left out of account, because the 
current is a maximum. 

The numerical value of the lines of force which go 
through the coil when inclined at an angle 6 to the 
meridian, is the horizontal intensity multiplied by the 
area of the coil and by sin 6; say nH-n-a 2 sin 0, where 
H = -1794, a =15, and n = 3QO. The electromotive 
force at any instant is the rate at which this quantity 
increases or diminishes ; that is, nH.7ra 2 cos 6 . w, if w 
denote the angular velocity. At the instant of passing 
the meridian cos 9 is 1, and the electromotive force is 
7iH7ra 2 w. With 10 revolutions per second the value of o> 
is 27r x 10. 

Hence the electromotive force is 

1794 x (3142) 2 x 225 x 20 x 300 = 2-39 x 10 6 . 

This is about of a volt. 
42 

190. To investigate the magnitudes of units of length, 
mass, and time which will fulfil the three following 
conditions : 

1. The acceleration due to the attraction of unit mass 
at unit distance shall be unity. 

2. The electrostatic units shall be equal to the electro- 
magnetic units. 



186 UNITS AND PHYSICAL CONSTANTS. [CHAP. 

3. The density of water at 4 C. shall be unity. 

Let the 3 units required be equal respectively to L 
centinis., M grammes, and T seconds. 

We have in C.G.S. measure, for the acceleration due 
to attraction ( 72), 

acceleration = C , . mass a where C = 6 '48 x 10~ 8 
(distance) 2 

and in the new system we are to have 
acceleration = mass 



(distance) 2 ' 
Hence, by division, 

acceleration in C.G.S. units 

acceleration in new units 

_ Q mass in C.G.S. units (distance in new units) 2 
mass in new units " (distance in C.G.S. units) 2 ' 

that is, ^(f, 

This equation expresses the first of the three conditions. 
The equation = v expresses the second, v denoting 

3 x 10 10 . 

The equation M = L 3 expresses the third. 
Substituting L 3 for M in the first equation, we find 

T = A /~. Hence, from the second equation, 

\ ^ 



and from the third, 



XL] ELECTRICITY. 187 

Introducing the actual values of C and v y we have 
approximately 

T = 3928, L= 1-178 x 10 14 , M = 1-63x10*2; 
that is to say, 

The new unit of time will be about l h 5 J m ; 

The new unit of length will be about 118 thousand 

earth quadrants ; 
The new unit of mass will be about 2 '6 6 x 10 14 times 

the earth's mass. 

Electrodynamics. 

191. Ampere's formula for the repulsion between two 
elements of currents, when expressed in electromagnetic 
units, is 

cc' ds . ds' /c . . / /) /> 

- - -- (2 sm a sin a cos - cos a cos a ), 

where c, c' denote the strengths of the two currents ; 
ds, ds' the lengths of the two elements ; 

a, a! the angles which the elements make with 

the line joining them ; 
r the length of this joining line ; 
the angle between the plane, of r, ds } and 

the plane of r, ds'. 

For two parallel currents, one of which is of infinite 
length, and the other of length Z, the formula gives by 
integration an attraction or repulsion, 



where D denotes the perpendicular distance between the 
currents. 



188 UNITS AND PHYSICAL CONSTANTS. [CHAP. xi. 

Exa/mple. 

Find the attraction between two parallel wires a metre 
long and a centim. apart when a current of is passing 

through each. 

Here the attraction will be sensibly the same as if 
one of the wires were indefinitely increased in length, 
and will be 



200/1 



, 



that is, each wire will be attracted or repelled with a force 
of 2 dynes, according as the directions of the currents are 
the same or opposite. 



189 



OMISSION (TO BE ADDED TO 63, p. 61). 

ACCORDING to experiments by Quincke (Berlin Transac- 
tions, April 5, 1885) the following are the compressions 
due to the pressure of one atmosphere. They are ex- 
pressed in millionths of the original volume : 



Glycerine, 25'24 

Rape oil (rttbol), 48 '02 

Almond oil, 48'21 

Olive oil, 48-59 

Water, 

Bisulphide of carbon, .... 

Oil of turpentine, 

Benzol from benzoic acid 

Benzol, 

Petroleum, 

Alcohol, 

Ether, ...115'57 



at C. 


it t C. 


t. 


25-24 


25-10 


19-00 


48-02 


58-18 


17-80 


48-21 


56-30 


19-68 


48-59 


61-74 


18-3 


50-30 


45-63 


22-93 


53-92 


63-78 


17-00 


58-17 


77-93 


18-56 





66-10 


16-78 





62-84 


16-08 


64-99 


74-50 


19-23 


82-82 


95-95 


17-51 


115-57 


147-72 


21-36 



CORRECTION (p. 84). 

BEXOIT'S results on refraction of air will not appear in 
vol. v., but in a later volume. 



190 



SUGGESTION FOR WRITING DECIMAL 
MULTIPLES AND SUBMULTIPLES. 

PROFESSOR NEWCOMB has suggested, as a possible improve- 
ment in future editions of this work, the employment of 
powers of 1000 instead of powers of 10 as factors (a plan 
which corresponds with the usual division of digits into 
periods of 3 each), and the employment of the letter m in 
this connection to denote 1000. 

Thus, instead of 1'226 x 10 5 , we should write 122-6 m. 
l-006xl0 7 , 10-06 m 2 . 

000 000 9, -9 m~~. 

The plan appears to possess some advantages ; and if the 
symbol m for 1000 is not sufficiently self-explanatory, we 
might write 122'6 x 10 3 , 10'06 x 10 6 , '9 x 10~ 6 . We 
place the suggestion on record here that it may not be 
overlooked. 



191 




APPENDIX. 



First Report of the Committee for the Selection and Nomenclature 
of Dynamical and Electrical Units, the Committee consisting of 
SIR W. THOMSON, F.R.S., PROFESSOR G. C. FOSTER, F.R.S., 
PROFESSOR J. C. MAXWELL, F.R.S., MR. G. J. STONEY, 
F. R.S.,* PROFESSOR FLEEMING JENKIN, F.R.S., DR. SIEMENS, 
F.R.S., MR. F. J. BRAMWELL, F.R.S., and PROFESSOR 
EVERETT (Reporter). 

WE consider that the most urgent portion of the task intrusted 
to us is that which concerns the selection and nomenclature of 
units of force and energy ; and under this head we are prepared 
to offer a definite recommendation. 

A more extensive and difficult part of our duty is the selection 
and nomenclature of electrical and magnetic units. Under this 
head we are prepared with a definite recommendation as regards 
selection, but with only an interim recommendation as regards 
nomenclature. 

Up to the present time it has been necessary for every person 
who wishes to specify a magnitude in what is called " absolute " 
measure, to mention the three fundamental units of mass, length, 
and time which he has chosen as the basis of his system. This 
necessity will be obviated if one definite selection of three funda- 
mental units be made once for all, and accepted by the general 
consent of scientific men. We are strongly of opinion that such 
a selection ought at once to be made, and to be so made that 
there will be no subsequent necessity for amending it. 

We think that, in the selection of each kind of derived unit, all 
arbitrary multiplications and divisions by powers of ten, or other 
factors, must be rigorously avoided, and the whole system of 

* Mr. Stoney objected to the selection of the centimetre as the unit of length. 



192 APPENDIX. 

fundamental units of force, work, electrostatic, and electromag- 
netic elements must be fixed at one common level that level, 
namely, which is determined by direct derivation from the three 
fundamental units once for all selected. 

The carrying out of this resolution involves the adoption of 
some units which are excessively large or excessively small in 
comparison with the magnitudes which occur in practice ; but a 
remedy for this inconvenience is provided by a method of denoting 
decimal multiples and sub-multiples, which has already been 
extensively adopted, and which we desire to recommend for 
general use. 

On the initial question of the particular units of mass, length, 
and time to be recommended as the basis of the whole system, a 
protracted discussion has been carried on, the principal point 
discussed being the claims of the gramme, the metre, and the 
second, as against the gramme, the centimetre, and the second, 
the former combination having an advantage as regards the 
simplicity of the name metre, while the latter combination has 
the advantage of making the unit of mass practically identical 
with the mass of unit-volume of water in other words, of making 
the value of the density of water practically equal to unity. We 
are now all but unanimous in regarding this latter element of 
simplicity as the more important of the two ; and in support of 
this view we desire to quote the authority of Sir W. Thomson, 
who has for a long time insisted very strongly upon the necessity 
of employing units which conform to this condition. 

We accordingly recommend the general adoption of the Centi- 
metre, the Gramme, and the Second as the three fundamental 
units ; and until such time as special names shall be appropriated 
to the units of electrical and magnetic magnitude hence derived, 
we recommend that they be distinguished from "absolute" units 
otherwise derived, by the letters "C.G.S. " prefixed, these being 
the initial letters of the names of the three fundamental units. 

Special names, if short and suitable, would, in the opinion of a 
majority of us, be better than the provisional designations "C.G.S. 
unit of . . . ." Several lists of names have already been 
suggested ; and attentive consideration will be given to any further 



APPENDIX. 193 

suggestions which we may receive from persons interested in 
electrical nomenclature. 

The "ohm," as represented by the original standard coil, is 
approximately 10 9 C.G.S. units of resistance; the "volt" is 
approximately 10 8 C.G.S. units of electromotive force ; and the 

"farad" is approximately - of the C.G.S. unit of capacity. 

For the expression of high decimal multiples and sub-multiples, 
we recommend the system introduced by Mr. Stoney, a system 
which has already been extensively employed for electrical pur- 
poses. It consists in denoting the exponent of the power of 
10, which serves as multiplier, by an appended cardinal num- 
ber, if the exponent be positive, and by a prefixed ordinal number 
if the exponent be negative. 

Thus 10 9 grammes constitute a gramme-nine; - Q of a gramme 

constitutes a ninth-gramme; the approximate length of a quadrant 
of one of the earth's meridians is a metre-seven, or a centimetre- 
nine. 

For multiplication or division by a million, the prefixes mega * 
and micro may conveniently be employed, according to the present 
custom of electricians. Thus the megohm is a million ohms, and 
the microfarad is the millionth part of a farad. The prefix mega 
is equivalent to the affix six. The prefix micro is equivalent to 
the prefix sixth. 

The prefixes kilo, hecto, deca, deci, centi, milli can also be em- 
ployed in their usual senses before all new names of units. 

As regards the name to be given to the C.G.S. unit of force, we 
recommend that it be a derivative of the Greek 8tva.fjus. The form 
dynamy appears to be the most satisfactory to etymologists. 
Dynam is equally intelligible, but awkward in sound to English 
ears. The shorter form, dyne, though not fashioned according to 
strict rules of etymology, will probably be generally preferred in 
this country. Bearing in mind that it is desirable to construct a 
system with a view to its becoming international, we think that 

* Before a vowel, either meg or megal, as euphony may suggest, may be 
employed instead of mega. 



194 APPENDIX. 

the termination of the word should for the present be left an open 
question. But we would earnestly request that, whichever form of 
the word be employed, its meaning be strictly limited to the unit of 
force of the C.G.S. system that is to say, the force which, acting 
upon a gramme of matter for a second, generates a velocity of a 
centimetre per second. 

The C.G.S. unit of work is the work done by this force working 
through a centimetre; and we propose to denote it by some deriva- 
tive of the Greek Zpyov. The forms ergon, ergal, and erg have 
been suggested; but the second of these has been used in a 
different sense by Clausius. In this case also we propose, for 
the present, to leave the termination unsettled ; and we request 
that the word ergon, or erg, be strictly limited to the C.G.S. unit 
of work, or what is, for purposes of measurement, equivalent to 
this, the C.G.S. unit of energy, energy being measured by the 
amount of work which it represents. 

The C.G.S. unit of power is the power of doing work at the rate 
of one erg per second ; and the power of an engine, under given 
conditions of working, can be specified in ergs per second, 

For rough comparison with the vulgar (and variable) units 
based on terrestrial gravitation, the following statement will be 
useful : 

The weight of a gramme, at any part of the earth's surface, is 
about 980 dynes, or rather less than a kilodyne. 

The weight of a kilogramme is rather less than a megadyne, being 
about 980,000 dynes. 

Conversely, the dyne is about 1 '02 times the weight of a milli- 
gramme at any part of the earth's surface ; and the megadyne is 
about 1 '02 times the weight of a kilogramme. 

The kilogrammetre is rather less than the ergon-eight, being 
about 98 million ergs. 

The gramme-centimetre is rather less than the kilerg, being 
about 980 ergs. 

For exact comparison, the value of g (the acceleration of a body 
falling in vacuo) at the station considered must of course be 
known. In the above comparison it is taken as 980 C.G.S. units 
of acceleration. 



APPENDIX. 195 

One Iwrse-power is about three quarters of an erg-ten per second. 
More nearly, it is 7 '46 erg -nines per second, and one force-de-cheval 
is 7*36 erg-nines per second. 

The mechanical equivalent of one gramme-degree (Centigrade) 
of heat is 41'6 megalergs, or 41,600,000 ergs. 



Second Report of the Committee for the Selection and Nomenclature 
of Dynamical and Electrical Units, the Committee consisting of 
PROFESSOR SIR W. THOMSON, F.R.S., PROFESSOR G. C. 
FOSTER, F.R.S., PROFESSOR J. CLERK MAXWELL, F.R.S., 
G. J. STONEY, F.R.S., PROFESSOR FLEEMING JENKIN, F.R.S., 
DR. C. W. SIEMENS, F.R.S., F. J. BRAMWELL, F.R.S., 
PROFESSOR W. G. ADAMS, F.R.S., PROFESSOR BALFOUR 
STEWART, F.R.S., and PROFESSOR EVERETT (Secretary). 

THE Committee on the Nomenclature of Dynamical and Electrical 
Units have circulated numerous copies of their last year's Report 
among scientific men both at home and abroad. 

They believe, however, that, in order to render their recom- 
mendations fully available for science teaching and scientific 
work, a full and popular exposition of the whole subject of 
physical units is necessary, together with a collection of examples 
(tabular and otherwise) illustrating the application of systematic 
units to a variety of physical measurements. Students usually 
find peculiar difficulty in questions relating to units ; and even the 
experienced scientific calculator is glad to have before him con- 
crete examples with which to compare his own results, as a 
security against misapprehension or mistake. 

Some members of the Committee have been preparing a small 
volume of illustrations of the C.G.S. system [Centimetre-Gramme- 
Second system] intended to meet this want. 

[The first edition of the present work is the volume of illustra- 
tions here referred to.] 



196 



INDEX. 



The numbers refer to the pages. 



Acceleration, 25. 
Acoustics, 70-74. 
Adiabatic compression, 125. 
Air, collected data for, 129. 

, density of, 43. 

, expansion of, 99. 

, specific heat of, 94, 123. 
, thermal conductivity of, 

108. 

Ampere as unit, 151-153. 
Ampere's formula, 187. 
Aqueous vapour, pressure of, 

100-102. 

, density of, 102. 

Astronomy, 65-69. 
Atmosphere, standard, 42, 43. 

, its density upwards, 47. 

Atomic weights, 180. 
Attraction, constant of, 67. 

at a point, 17. 

Angle, 16. 
, solid, 17. 

Barometer, correction for capil- 
larity, 51. 

Barometric measurements of 
heights, 47. 

pressure, 42. 

Batteries, 166-168, 172, 181. 

Boiling points, 98. 

of water, 100-102. 



Boyle's law, departures from, 99. 
Bullet, melted by impact, 31. 

Candle, standard, 86. 

Capacity, electrical, 141-143. 

, specific inductive, 147-150. 

, thermal, 87-95. 

Capillarity, 49-51. 

Carcel, 86. 

Cells, 166-168, 172, 181. 

Centimetre, reason for selecting, 
23, 192. 

Centre of attraction, strength 
of, 17. 

Centrifugal force, 32. 

at equator, 34. 

C.G.S. system, 23, 192. 

Change of volume in evapora- 
tion, 97. 

in melting, 96, 97. 

Change-ratio, 9. 

Chemical action, heat of, 122. 

equivalents, 180. 

Clark's standard cell, 168. 

Cobalt, magnetization of, 136. 

Coil, revolving, 185. 

Combination, heat of, 122, 181. 

Combustion, heat of, 122. 

Common scale needed, 22. 
omparisoiiof standards (French 
and English), 1, 2. 



INDEX. 



197 



Compressibility of liquids, 60, 
61, 189. 

of solids, 61-63. 

Compression, adiabatic, 125. 
Conductivity (thermal) defined, 

103. 
, thermometric and calori- 

metric, 105. 

of air, 108. 

of liquids, 116, 117. 

of various solids, 109-116. 

Congress of electricians, 153. 
Contact electricity, 168-172. 
Cooling, 117-120. 
Current, heat generated by, 

143, 166. 

, unit of, 141, 142, 151, 153. 

Curvature, dimensions of, 17, 18. 

Daniell's cell, 166, 167, 173, 181. 

Day, sidereal, 66. 

Decimal multiples, 24, 190, 193. 

Declination, magnetic, at Green- 
wich, 138. 

Densities, table of, 40. 

of gases, 44. 

of water, 38-39. 

Density as a fundamental unit, 
146. 

Derived units, 5, 6. 

Dew-point from wet and dry 
bulb, 102. 

Diamagnetic substances, 133. 

Diamond, specific heat of, 90. 

Diffusion, coefficient of, 105-108. 

Diffusivity (thermal), 105. 

Dimensional equations, 9, 34-37. 

Dimensions, 7-9, 34-37. 

Dip at Greenwich, 138. 

Dispersive powers of gases, 83- 
85. 

of solids and liquids, 

77-82. 

Diversity of scales, 22. 

"Division," extended sense of, 
10. 

Double refraction, 81. 



Dynamics, 15-17. 
Dyne, 27, 193. 

Earth as a magnet, 136. 
, size, figure, and mass of, 

65. 

Elasticity, 52-64. 
, effected by heat of com- 
pression, 127. 
Electric units, tables of their 

dimensions, 143, 146. 
Electricity, 140-148. 
Electrochemical equivalents, 

180. 

Electrodynamics, 187. 
Electromagnetic units, 142. 
Electromotive force, 166-172, 

180-182. 

Electrostatic units, 140. 
Emission of heat, 117-120. 
Energy, 29. 

, dimensions of, 16. 
Equations, dimensional, 9, 34-37. 

, physical, 12. 
Equivalent, mechanical, of heat, 

120. 
Equivalents, electrochemical, 

180. 

Erg, 29, 194. 
Evaporation, change of volume 

in, 97. 

Examples in electricity, 182-1 86. 
in theory of units, 12-15, 



34-37. 

in magnetism, 134-137. 

Expansion of gases, 99. 

of mercury, 128, 129. 

of various substances, 128, 

129. 

Extended sense of "multiplica- 
tion" and "division," 10. 

Farad, 151-153. 

compared with earth, 183. 
Field, intensity of, 131. 
Films, tension in, 49, 50. 
, thickness of, 50, 51. 






198 



INDEX. 



Foot-pound and foot-poundal, 

Force, 27. 

, dimensions of, 15. 

at a point, 17. 

, various units of, 4. 

Freezing-point, change with 
pressure, 124. 

Frequencies of luminous vibra- 
tions, 77. 

Fundamental units, 6. 

, choice of, 19. 

reduced to two, 68. 

Gases, densities of, 44. 

, expansion of, 99. 

, indices of refraction of, 82. 

, inductive capacities of, 

151. 

, two specific heats of, 123. 
Gauss's expression for magnetic 

potential, 138. 

pound-magnet, 134. 

units of intensity, 137. 

Geometrical quantities, dimen- 
sions of, 15-18. 
Gottingen, total intensity at, 

137. 
Gramme-degree (unit of heat), 

88. 

Gravitation in astronomy, 67. 
Gravitation measure of force 

and work, 28, 30. 
Gravity, terrestrial, 25-27. 
Greenwich, magnetic elements 

at, 138. 
Grove's cell, 167, 172, 181. 

Heat, 87-130. 

generated by current, 143, 

166. 
, mechanical equivalent of, 

120. 

of combination, 122, 181. 

of compression, 125. 

, unit of, 87, 88. 

, various units of, 3. 



Height, measured by barometer, 

47. 

Homogeneous atmosphere, 45-47. 
Horse-power, 30. 
Hydrostatics, 38-51. 
Hypsometric table of boiling 

points, 100. 

Ice, specific gravity of, 96, 125. 
, specific heat of, 92. 



, electrical resistance of, 
163. 

Indices of refraction, 77-85. 

related to induc- 
tive capacities, 147, 148. 

Inductive capacity, 147-150. 

Induction, magnetic, coefficient 
of, 133. 

Insulators, resistance of, 164, 
165. 

Interdiffusion, 106-108. 

Joule's equivalent, 120. 

Kilogramme and pound, 2. 
Kinetic energy, 29. 
Kupffer's determination of den- 
sity of water, 38. 

Large numbers, mode of expres- 
sing, 24, 190, 193. 
Latent heats, 95-98. 
Latimer Clark's cell, 168. 
Light, 75-86. 

, velocity of, 75, 76. 

, wave-lengths of, 76. 



Magnetic elements at Green- 
wich, 138. 

susceptibility, 133. 
- units, 131, 132. 
Magnetism, 131-139. 

, terrestrial, 136-138. 
Magneto-optic rotation, 139. 
Magnetization, intensity of, 

132, 133. 
Mass, standards of, 20. 



INDEX. 



199 



Mechanical equivalent of heat, 

120. 
quantities, dimensions of, 

15. 

units, 27. 

Mega, as prefix, 42, 193. 
Melting points, 95-97. 
Metre and yard, 1. 
Micro as prefix, 193. 
Microfarad, 151. 
Moment of couple, 16. 
of inertia, 16. 

of magnet, 132. 

of momentum, 16. 
Momentum, 15. 
Moon, 66. 

' 'Multiplication, "extended sense 
of, 10. 

Neutral point (thermoelectric), 

175. 
Newcomb on decimal multiples, 

190. 

Nickel, magnetization of, 136. 
Numerical value, 5. 

Ohm as unit, 151-154. 

earth quadrant per second, 

155. 

, "legal," 153. 

Optics, 75-86. 

Paramagnetic substances, 133. 

Pendulum, seconds', 25, 26. 

" Per," meaning of, 10. 

Physical deductions from di- 
mensions, 34-37. 

Platinoid, 160. 

Platinum, specific heat of, 90. 

Poisson's ratio, 62. 

Potential, electric, 140. 

, magnetic, 131. 

Poundal, 28. 

Powers of ten as factors, 24, 
190, 193. 

Pressure, dimensions of, 17. 

of liquid columns, 42. 



Pressure, various units of, 3, 4. 
Pressures of vapours, 101. 
Pressure- height, 46. 

Quantity of electricity, 140, 142. 

Radian, 16. 

Radiation, 117-120. 

Ratios of two sets of electric 
units, 143. 

Refraction, indices of, 77-85. 

Reports of Units Committee, 
191-195. 

Resilience, 54. 

affected by heat of com- 
pression, 126. 

Resistance, electrical, 158-166. 

of a cell, 161, 183. 

of wires, 165, 166. 

Rigidity, simple, 55. 

Rotating coil, 185. 

Saturation, magnetic, 133-136. 
Shear, 55-58. 
Shearing stress, 58-60. 
Siemens' unit, 152-154. 
Soap films, 50. 
Sound, faintest, 74. 

, velocity of, 70-73. 

Spark, length of, 155-157. 
Specific gravities, 40. 

, heat, 88-95. 

, two, of gases, 123. 

, inductive capacity, 147- 

150. 

Spring balance, 31. 
Standards, French and English, 

1,2. 

of length, 21. 

of mass, 20. 

of time, 21. 

Steam, pressure and density of, 

100-102. 
, total and latent heat of, 

98. 
Stoney's nomenclature for 

multiples, 193. 



200 



INDEX. 



Strain, 52, 53, 55-58. 

, dimensions of, 53. 
Stress, 52-54, 58-60. 

, dimensions of, 54. 

Strings, musical, 73. 
Sun's distance and parallax, 66. 
Supplemental section on dimen- 
sions, 34-37. 

Surface-conduction, 117-120. 
Surface-tension, 49, 50. 

Telegraphic cables, resistance 
of, 160. 

Tenacities, table of, 64. 

Tensions of liquid surfaces, 49, 
50. 

Thermodynamics, 120-128. 

Thermoelectricity, 172-178. 

Time, standard of, 21. 

Tortuosity, 17. 

Two fundamental units suffici- 
ent, 68. 

Unit, 5. 

Units, derived, 5, 6. 

, dimensions of, 7-9. 

, special problems on, 69, 

185. 

Vapours, pressure of, 101. 



Velocity, 6, 9. 

of light, 75, 76. 

of sound, 70-73. 

, various 'units of, 2. 
Vibrations per second of light, 

77. 

Volt, 151-153. 
Volume, by weighing in water, 

40. 

of a gramme of gas, 44. 

, unit of, 5. 

Volume resilience, 55, 60-63, 189. 

Water, compressibility of, 60, 
61, 189. 
, density of, 38, 39. 



, expansion of, 39. 

, specific heat of, 87, 88. 

, weighing in, 40. | 

Watt (rate of working), 4, 30. 
Weight, force, and mass, 27, 28. 

, standards of, 20-2. 
Wires (Imperial gauge), 165, 166. 
Work, 29, 30, 3, 4. 

, dimensions of, 16. 

done by current, 143. 



Working, rate of, 30, 3, 4. 

Year, sidereal and tropical, 6 
Young's modulus, 55. S . 



PRINTED BY ROBERT MACLEHOSE, UNIVERSITY PRESS, GLASGOW. 



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